Research ArticleMoving Clusters within a Memetic Algorithm forGraph Partitioning

Inwook Hwang1 Yong-Hyuk Kim2 and Yourim Yoon3

1Technical Laboratory Atto Research 225-18 Pangyoyeok-ro Bundang-gu Seongnam-si Gyeonggi-do 463-400 Republic of Korea2Department of Computer Science amp Engineering Kwangwoon University 20 Kwangwoon-ro Nowon-guSeoul 139-701 Republic of Korea3Department of Computer Engineering College of Information Technology Gachon University 1342 SeongnamdaeroSujeong-gu Seongnam-si Gyeonggi-do 461-701 Republic of Korea

Correspondence should be addressed to Yourim Yoon yryoongachonackr

Received 23 September 2014 Accepted 7 January 2015

Academic Editor John Gunnar Carlsson

Copyright copy 2015 Inwook Hwang et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Most memetic algorithms (MAs) for graph partitioning reduce the cut size of partitions using iterative improvement But this localprocess considers one vertex at a time and fails to move clusters between subsets when the movement of any single vertex increasescut size even though moving the whole cluster would reduce it A new heuristic identifies clusters from the population of locallyoptimized random partitions that must anyway be created to seed the MA and as the MA runs it makes beneficial cluster movesResults on standard benchmark graphs show significant reductions in cut size in some cases improving on the best result in theliterature

1 Introduction

Consider an unweighted undirected graph 119866 = (119881 119864) where119881 is a set of 119899 vertices and 119864 is the set of edges that connectthem A 119896-way partition 119875

1 1198752 119875

119896 of the graph 119866 is a

partitioning of the vertex set 119881 into 119896 disjoint subsets Apartition is said to be balanced if the difference in size betweenthe largest and the smallest subset is at most 1 that is for all1 le 119894 119895 le 119896 ||119875

119894| minus |119875119895|| le 1 The cut size of a partition is

defined to be the number of edges connecting vertices indifferent subsets of the partitionThe 119896-way graph partitioningproblem is the problem of finding a balanced 119896-way partitionwith the minimum cut size If 119896 = 2 it can be calledbipartitioning and if 119896 gt 2 multiway partitioning Theseproblems arise in applications such as sparsematrix factoriza-tion network partitioning layout and floor planning circuitplacement social network analysis and software-definednetworking [1 2]

For general graphs partitioning is known to be NP-hard[3] Bui and Jones [4] have shown that even finding goodapproximate solutions is also NP-hard

Therefore many heuristic methods have been proposedsome of them work well but they cannot of course guaranteeoptimality The simplest heuristic is iterative improvementpartitioning (IIP) [5 6] exemplified by the Kernighan-Lin (KL) [7] and the Fiduccia-Mattheyses (FM) algorithms[8] but these algorithms only produce solutions which areapproximations to local optima however this limitation canbe overcome by hybridizing them with metaheuristics suchas simulated annealing [9] genetic algorithms (GAs) [10]tabu search [11 12] or ant colony optimization [13] Recentlya number of techniques based on GAs have achieved notableresults for 119896 = 2 [14ndash19] and 119896 gt 2 [20ndash26] Kim et al [27]have surveyed this work

The use of IIP for local optimization of partitioning pro-duced by a GA becomes less effective as the graph becomeslargerWewill show that this is because IIP often fails tomovedensely interconnected subgraphs called clusters betweenpartitions and hence fails to find partitions with small cutsizes

The goal of the work reported in this paper is to over-come the barriers to effective search which are presented

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 238529 10 pageshttpdxdoiorg1011552015238529

2 Mathematical Problems in Engineering

by clusters by modifying the GA so that it contributes tomove clusters appropriatelyWe present a memetic algorithm(MA) which is a GA combined with local optimizationin which a heuristic finds clusters in some of the positionsin each generation by examining population of individualseach of which represents a position rather than trying toidentify them directly from a single graph It moves someof these clusters This heuristic supplements the well-knownability of MAs to provide attractive initial points for localoptimization Experimental results show that this approachcan substantially improve the performance of an MA Thecontributions of this work are summarized as follows

(i) We provide a detailed explanation of the difficultyof moving clusters in graph partitioning and provideexperimental results quantifying the impact of clus-ters on the search for partitions with a small cut size

(ii) We present a heuristic for detecting and movingclusters which is based on a new population-basedmeasure of the distance between vertices called genicdistance

(iii) We show that this heuristic substantially improves theability of an MA to find good partitions

The remainder of this paper is organized as follows InSection 2 we briefly introduce IIP algorithms and the testgraphs used in our experiments In Section 3 we investigatethe difficulty of moving clusters in graph partitioning InSection 4we describe our new cluster-handling heuristic andan MA that uses this heuristic is described in Section 5 InSection 6 we present experimental results and draw conclu-sions in Section 7

2 Preliminaries

21 Iterative Improvement Algorithms in Bipartitioning Itera-tive improvement partitioning starts with a randompartitionThis is refined in a series of passes At the start of each passall the vertices are free to move between subsets IIP selectsvertices and moves them but each vertex is only moved onceduring a pass At the end of the pass the best partition foundduring the pass is identified and used as the input to the nextpass Passes continue until there is no further improvement

There are a number of IIP algorithms of which KL [7]is often considered to be the first reasonable heuristic forbipartitioning In KL the movement of vertices during a passis restricted to the swapping of a pair of vertices betweensubsets

Let 119860 119861 be a partition of 119881 into two subsets 119860 and 119861We define the gain 119892V associated with a vertex V to be thereduction in cut size obtained bymoving V to the other subsetBy extension the gain 119892(119886 119887) obtained by swapping vertices119886 isin 119860 and 119887 isin 119861 can be expressed as follows

119892 (119886 119887) = 119892119886+ 119892119887minus 2120575 (119886 119887) (1)

where

120575 (119886 119887) = 1 if (119886 119887) isin 1198640 otherwise

(2)

KL selects the pair (119886 119887) with the highest value of 119892(119886 119887)and effects the exchange The vertices 119886 and 119887 are notconsidered again during the current pass A sequence of pairs(1198861 1198871) (1198862 1198872) (119886

1198992minus1 1198871198992minus1

) are selected in this wayThe algorithm chooses 119897 that maximizes sum119897

119894=1119892(119886119894 119887119894) and

exchanges 1198861 1198862 119886

119897 and 119887

1 1198872 119887

119897 KL performs

further passes until no improvement is possibleFM is another widely used IIP algorithm which is similar

to KL except that it only moves one vertex at a time Thismakes FM faster than KL with little loss in partition qualitySeveral variants of KL and FM exist [15 28 29]

22 Local Optimization Algorithms for Multiway PartitioningThere are three main schemes for multiway partitioningwhich are developments of the recursive pair-wise and directapproaches [21] to bipartitioningThe recursive KL algorithmbisects the graph recursively until there are 119896 subsets Thepair-wise KL [7] starts with an arbitrary 119896-way partition Itpicks two subsets at a time from the 119896 subsets and performsbipartitioning to reduce the cut size between those pairs San-chis [30] extended the FMalgorithm tomultiway partitioningand showed that the direct method performed better thanrecursion The extended algorithm considers moving eachvertex from its current subset to every other subset Toperform local optimization in the proposedMA formultiwaypartitioning we use a variant of this algorithm called EFM(extended FM) [21] The time complexity of EFM is 119874(119896|119864|)

23 Local Search in Memetic Algorithms It is already clearthat combining a GA with local optimization algorithmsis an effective approach to the graph partitioning problem[15] Some authors have explored fast but weak local opti-mization algorithms For example [31 32] 2-opt was usedto relocate border vertices which are those with edges thatconnect to vertices in other subsets Bui and Moon [10]obtained better results with KL by allowing only a single passwhile restricting the number of vertices to be swappedConversely other authors have reported notable improve-ments by enhancing local optimization algorithms For bipar-titioning Kim and Moon [15] suggested a new KL-basedlocal optimization algorithm formulated using a new typeof gain called lock gain which only takes into account theedges that connect a vertex to the vertices that have alreadybeen moved Combined with a GA this algorithm obtainedimpressive results on most benchmark graphs For multi-way partitioning the combination of MAs with specializedlocal optimization algorithms showed good results [20 21]Steenbeek et al [18] proposed what they called a clusterenhancement heuristic which they combined with an MAand reported successful results Their MA uses a vertexswap heuristic to identify clusters The MA only handles themoving of clusters between subsets

24 Test Graphs We tested our MA on Johnsonrsquos benchmarkgraphs [9] which have been widely used in other studies [1011 14ndash17 20 21 23 33ndash36] They are composed of 8 random

Mathematical Problems in Engineering 3

Cluster

3

4

1

2

5 edges connecting subsets

(a) Before moving a cluster

Cluster

3

4

1

2

1 edge connecting subsets

(b) After moving a cluster

Figure 1 An example of cluster moving in which the cut size of apartition is reduced by 4

graphs G119899119889 and 8 random geometric graphs U119899119889 The twodifferent classes of graphs are briefly described below

(i) G119899119889 a random graph on 119899 vertices with an indepen-dent probability 119901 that any two vertices are connectedby an edge The probability 119901 is biased so that theexpected vertex degree 119901(119899 minus 1) is 119889

(ii) U119899119889 a random geometric graph on 119899 vertices that liein the unit square and whose coordinates are chosenuniformly from the unit interval Every pair of ver-tices separated by a distance of 119905 or less is connectedby an edge The expected degree of a vertex is 1198991205871199052

3 Difficulty of Moving Clusters

Suppose that the cluster shown in Figure 1(a) is involved ina bipartitioning problem The four vertices in this cluster arefully interconnected and they all belong to the same subsetMoving this cluster to the other subset across the dottedline in Figure 1(b) will reduce the cut size of the partition by4 However there is no motivation to move any single vertexbecause they all have negative gain the gains of V

1 V2 V3 and

V4are minus1 minus4 minus2 and minus1 respectivelyThis example illustrates

how IIP algorithms may miss a significant reduction incut size that could be achieved by moving several verticestogether

The baleful effect of clusters on local search algorithmstrying to solve the graph partitioning problemmotivated thisstudy Kim [37 38] indicated that graph partitioning is hardprimarily due to the difficulty of moving clusters Dutt andDeng [39 40] have also observed that an IIP method appliedto circuit partitioning can fail because of the difficulty ofdealing with clusters that straddle subsets

31 Experimental Support We designed experiments toquantify the effect of clusters on IIP algorithms representedby the KL algorithm Using the cluster detection method tobe described in Section 41 we find clusters in the graph andselect one randomly We then take a locally optimum bipar-tition 119904 obtained by KL and move the selected cluster to theother subset creating a perturbed partition 119905cluster ApplyingKL to 119905cluster we obtain a new local optimum 119906cluster

Table 1 Probability that KL fails to return vertices moved from onesubset of a partition to the other when the vertices are in cluster(119875cluster) or chosen at random (119875random) over 1000 runs

Graph 119875cluster () 119875random () Average number of verticesmoved (119902)

G50025 1690 250 641

G50005 570 130 550

G50010 190 030 505

G50020 050 010 478

G100025 1280 190 589

G100005 290 080 561

G100010 060 020 467

G100020 030 000 491

U50005 3910 100 741

U50010 2620 050 1205

U50020 860 020 1410

U50040 1040 040 4936

U100005 3790 080 781

U100010 2850 060 1247

U100020 1160 010 2219

U100040 910 040 3588

Assuming that 119902 the number of vertices in the clusters issmall 119906cluster can be expected to match 119904 if KL is successfulin moving the cluster back However if KL fails to returnthe perturbed cluster to its original subset the cut size of thepartition may increase Repeating this experiment we derive119875cluster as an approximation of the probability that the cut sizeof 119906cluster is larger than that of 119904

For comparison we perturbed 119902 vertices randomlyselected within a locally optimized partition bymoving themto the other subset We call this partition 119905random We applythe KL algorithm to 119905random and then obtain a new locallyoptimum partition 119906random Repeating this experiment wederive 119875random as an approximation of the probability that thecut size of 119906random is larger than that of 119904

Table 1 shows the values of 119875cluster and 119875random for 16benchmark graphs We see that 119875cluster is always larger than119875random as we would expect We notice that the gap between119875cluster and 119875random is larger on the geometric graphs (U119899119889)than on the random graphs (G119899119889)

4 Cluster-Handling Heuristic

Cluster analysis is a well-known problem for which plenty ofalgorithms exist many of which require a lot of computationThe insight that motivates our heuristic is that the applicationof a local optimization process such as IIP to a randomly par-titioned graph creates a modified partition in which clusterstend to be wholly allocated to one subset or another (and arethen difficult to move as we have already observed) A singlepartition of this sort is of little help in identifying clustersbecause the clusters are not separated at all within eachsubset but if we create many random partitions and optimize

4 Mathematical Problems in Engineering

Table 2 Truth table for Fact 1

119901 119902 119901 rarr 119902 119868(119901) 119868(119902)

True True True 1 1

True False False 1 0

False True True 0 1

False False True 0 0

them we can reasonably infer that vertices that find them-selves in the same subset in most of these partitions belong tothe same cluster We can make this inference in a structuredway using the ldquogenic distancerdquo metric that we will introduceThis approach to cluster analysis may seem indirect but it isefficient in the context of an evolutionary approach to graphpartitioning because the set of partitions required for findingclusters using genic distance is also the population which wemust create to be evolved by our MA

One way of dealing with clusters is to devise a local opti-mization heuristic that can identify clusters [18 19 38] How-ever this prevents us from building on previous studies of IIPalgorithms

Our approach is to add an additional heuristic to ourMA which finds and moves clusters The heuristic identifiesclusters in the population of partitions which have alreadybeen optimized locally It selects clusters with higher gainsandmoves them IIP local optimization is then applied again

41 Cluster Detection Let 119868(sdot) be an indicator function that is119868(true) = 1 and 119868(false) = 0 Then we can trivially establishthe following

Fact 1 If 119901 rarr 119902 is true then 119868(119901) le 119868(119902)

Proof From Table 2

Fact 2 119868(119901 or 119902) le 119868(119901) + 119868(119902)

Proof From Table 3

We now define a metric called genic distance which mea-sure the extent to which two vertices connected by an edgecan be considered to belong to the same cluster We denotethe genic distance of an edge V

119894 V119895 within a population

1199011 1199012 119901

119898 of locally optimized partitions as 119889

119892(V119894 V119895)

which can be expressed as follows

119889119892(V119894 V119895) =

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119895)) (3)

where 119892119894and 119892

119895are the genes corresponding to V

119894and V

119895

respectively The value of gene 119892119894(ie the partition number

that vertex V119894belongs to) in the 119897th individual is 119901

119897(119892119894) For

convenience we assume that each vertex has an edge thatconnects it to itself so that V V isin 119864 for each vertex V Thenthe following proposition holds

Proposition 1 For each population 119889119892becomes a pseudomet-

ric on 119881

Table 3 Truth table for Fact 2

119901 119902 119868(119901) 119868(119902) 119901 or 119902 119868(119901 or 119902) 119868(119901) + 119868(119902)True True 1 1 True 1 2

True False 1 0 True 1 1

False True 0 1 True 1 1

False False 0 0 False 0 0

Proof Since 119868(sdot) ge 0 119889119892(V119894 V119895) = sum

119898

119897=1119868(119901119897(119892119894) = 119901119897(119892119895)) ge 0

for each V119894 V119895 isin 119864 It is enough to show the following three

conditions

(i) 119889119892(V119894 V119894) = 0

119889119892(V119894 V119894) =

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119894))

=

119898

sum

119897=1

119868 (false)

= 0

(4)

(ii) Symmetry Let V119894 V119895 be in 119864

119889119892(V119894 V119895) =

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119895))

=

119898

sum

119897=1

119868 (119901119897(119892119895) = 119901119897(119892119894))

= 119889119892(V119895 V119894)

(5)

(iii) Triangle Inequality Consider each group of threeedges V

119894 V119895 V119894 V119896 V119896 V119895 isin 119864 If 119901

119897(119892119894) = 119901

119897(119892119896)

and119901119897(119892119896) = 119901119897(119892119895) then119901

119897(119892119894) = 119901119897(119892119895) for each 119897 By

contraposition if 119901119897(119892119894) = 119901119897(119892119895) then 119901

119897(119892119894) = 119901119897(119892119896)

or 119901119897(119892119896) = 119901119897(119892119895)

For each 119897

119868 (119901119897(119892119894) = 119901119897(119892119895))

le 119868 (119901119897(119892119894) = 119901119897(119892119896) or 119901119897(119892119896) = 119901119897(119892119895)) (∵ Fact 1)

le 119868 (119901119897(119892119894) = 119901119897(119892119896)) + 119868 (119901

119897(119892119896) = 119901119897(119892119895)) (∵ Fact 2)

(6)

By summing the above inequalities for all 119897

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119895))

le

119898

sum

119897=1

(119868 (119901119897(119892119894) = 119901119897(119892119896)) + 119868 (119901

119897(119892119896) = 119901119897(119892119895)))

=

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119896)) +

119898

sum

119897=1

119868 (119901119897(119892119896) = 119901119897(119892119895))

(7)

Mathematical Problems in Engineering 5

Therefore we have

119889119892(V119894 V119895) le 119889119892(V119894 V119896) + 119889119892(V119896 V119895) (8)

That is 119889119892satisfies the triangle inequality

Proposition 1 suggests that the measure 119889119892is reasonable

A pseudometric space is a generalization of a metric space inwhich points need not be distinguishable thus it is possiblethat 119889

119892(V119894 V119895) = 0 for some edge V

119894 V119895 with distinct vertices

V119894= V119895

Our heuristic detects clusters by collecting a number oflocal optima and computes genic distances for all the edges ineach graphThis takes119874(119898|119864|) time but this cost is negligiblesince this computation is a preprocess performed before theMA runs The heuristic temporarily eliminates edges withgenic distances that are greater than a threshold value 120579 Weset 120579 to be the smallest value that satisfies

10038161003816100381610038161003816119890 isin 119864 119889

119892(119890) le 120579

10038161003816100381610038161003816le 01 |119881| (9)

Each remaining connected component containingmore thanthree vertices is considered to be a cluster

Figure 2 shows how our heuristic detects clusters Fig-ure 2(a) shows an example graph with 11 vertices and 15edges Four individuals corresponding to locally optimizedpartitions from the population are shown in Figure 2(b)In Figure 2(c) each edge is labeled with its genic distanceIf the threshold value of genic distance is 1 then theedges with larger genic distances indicated by dotted lineare eliminated Then four connected components remainV3 V7 V8 V9 V2 V4 V6 V11 V1 V5 and V

10 The last two

of these connected components are considered too small tobe clusters Thus clusters V

3 V7 V8 V9 and V

2 V4 V6 V11

shaded in Figure 2(c) remain as candidates for moving

42 Cluster-Moving Scheme Our heuristic improves theoffspring of each generation after crossover by moving theclusters that were detected using the technique described inthe previous subsection To select the clusters to be movedand their target subsets we introduce ameasure called clustergain such that cg(119909 119886) is the reduction in the cut size of thepartition when all the vertices in cluster 119909 are moved to thesubset 119886 For example moving the cluster in Figure 1(a) to theother subset in the partition is associated with a cluster gainof 2

This cluster-moving scheme described in Algorithm 1is applied to each individual generated by crossover whichis a partition that may be unbalanced However cut sizeand cluster gain are well defined on unbalanced partitionsOur scheme does not consider moving every cluster in everypartition because we found that making all clusters movablecauses the premature convergence of the MA Thus ourheuristic selects119873 clusters at random as candidates for mov-ing In our experiments we set119873 to 5We compute the clustergain that results from moving each candidate cluster to eachof the other 119896 minus 1 subsets The candidate cluster 119909 anddestination subset 119886with the highest cluster gain are selectedAssume that cg(119909 119886) is positive all the vertices in cluster 119909

6

12

3 45

78

9 1011

(a) Example graph

1

1 1 1 1 1 1

1 113 5 7 8 11

1

1111110 0 0 0 0

10 0

0 0 0 0 0

1 1 1 1 10 00 0 0 0

0 0 0 010642 9

(b) Four locally optimized partitionsin the population (subsets 0 and 1)

00

00

0

00

11

1

23

03

3

(c) Two detected clusters

Figure 2 An example of cluster detection

are moved to subset 119886 and cluster 119909 is removed from theset of candidate clusters This process is repeated until nocandidates remain or no move yields a positive cluster gainNo attempt is made to balance the partition during cluster-moving this is performed later

5 Memetic Search Framework

CMPA (cluster-moving memetic partitioning algorithm) is amemetic algorithm that we have designed for graph parti-tioning In this MA an individual is a 119896-way partition Eachgene in an individual corresponds to a vertex and has a valuebetween 0 and 119896 minus 1 which indicates the subset to which thevertex belongs that is the 119894th gene 119892

119894= 119895 hArr V

119894isin 119875119895+1

It isa steady-state MA meaning that there is only one offspringfrom population in each generation Crossover is followed bya cluster-moving step and then local optimization

Algorithm 2 shows the processes that make up CMPAwhich we now describe in detail

(i) Initialization When the MA starts 119898 individuals(ie partitions) are created at random Then eachindividual is improved by local optimization We set119898 to be 30 for bipartitioning and 50 for multiwaypartitioning (with 119896 = 8 or 119896 = 32)

6 Mathematical Problems in Engineering

Select clusters 119878 larr 119873 clusters selected at randomdo

Calculate cg(119909 119886) for all 119909 isin 119878 1 le 119886 le 119896cglowast larr max

119909119886cg(119909 119886)

(119909 119886) larr argmax119909119886

cg(119909 119886)

if cglowast gt 0 then

Move cluster 119909 to partition 119886119878 larr 119878 119909

until (119878 = 0 or cglowast le 0)

Algorithm 1 Our cluster-moving scheme

Create initial population of fixed sizeApply local optimization to each member of populationCalculate genic distance from populationFind clusters and store their informationdo

Select 1199011198861199031198901198991199051and 119901119886119903119890119899119905

2from population

Normalization(1199011198861199031198901198991199051 119901119886119903119890119899119905

2)

offspring larr Crossover(1199011198861199031198901198991199051 119901119886119903119890119899119905

2)

Cluster-moving(offspring)Local-optimization(offspring)Replace(population offspring)

until (stopping condition)return the best solution

Algorithm 2 The process in CMPA

(ii) Selection We used the roulette-wheel-based propor-tional selection The probability that the best individ-ual is chosen was set to four times the probabilitythat the worst is chosen The fitness value 119891

119894of the

119894th individual is expressed as (119888119908minus 119888119894) + (119888119908minus 119888119887)3

where 119888119887 119888119908 and 119888

119894are the cut sizes of the partitions

corresponding to the best the worst and the 119894thindividual respectively

(iii) Normalization Laszewski [31] first used normaliza-tion to improve the performance of GA and its vari-ants have been suggested in [23 26 41 42]The parentindividuals are normalized before crossover followingLaszewski [31 33] The subset of one parent whichshares the largest number of vertices with subsetsof the other parent is selected and that subset isnumbered 0 This process is repeated incrementingthe index until all subsets are numbered

(iv) Crossover and Cluster Moving We used a standardfive-point crossover After crossover the cluster-han-dling heuristic described in Section 42 is applied tothe individual At this point individuals usually cor-respond to unbalanced partitionsWe select a randomlocation in the individual and adjust the values ofthe genes which are the subsets to which the corre-sponding vertices belong to the right of this location(in a typical circular string) until the partition is

balanced This is effectively a mutation effect and nofurther mutation was introduced

(v) Local Optimization KL [7] was used for the bisectionproblems and EFM (extended FM) [21] was used formultiway partitioning (119896 = 8 or 119896 = 32)

(vi) Replacement We used a replacement scheme due toBui and Moon [10] If an offspring is better than itscloser parent the MA replaces that parent If it is bet-ter than its other parent then that parent is replacedOtherwise it replaces the worst individual in thepopulation

(vii) Stopping Condition This is based on consecutive fail-ures to replace an individualrsquos parents Termination istriggered by consecutive failures 30 in bipartitioningand 50 in multiway partitioning

6 Experimental Results

We conducted experiments on 2-way 8-way and 32-way par-titioning Table 4 shows the performance of MA combinedwith KL (denoted KL-MA) and CMPA on bipartitioningTable 5 shows the performance of the genetic extended FMalgorithm (GEFM) [21] one of themost effective approachesand CMPA on 8-way partitioning and Table 6 gives theresults for 32-way partitioning CMPAuses a cluster-handling

Mathematical Problems in Engineering 7

Table 4 Comparison of KL-MA and CMPA on bipartitioning

Graph Best known1 KL-MA CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

G50025 49 49 5134 032 (595) 49 5122 034 (558)G50005 218 218 21865 059 (700) 218 21845 064 (682)G50010 626 626 62755 102 (681) 626 62746 110 (672)G50020 1744 1744 174583 193 (694) 1744 174562 209 (687)G100025 93 93 9726 096 (745) 93 9708 104 (722)G100005 445 445 45150 207 (884) 445 45092 222 (860)G100010 1362 1362 136668 407 (969) 1362 136618 445 (977)G100020 3382 3382 338561 742 (971) 3382 338526 813 (987)U50005 2 2 496 055 (793) 2 394 066 (803)U50010 26 26 2631 045 (459) 26 2604 057 (465)U50020 178 178 17806 042 (300) 178 17800 053 (271)U50040 412 412 41200 034 (190) 412 41200 037 (102)U100005 1 1 1167 171 (1077) 1 881 212 (1068)U100010 39 39 4627 151 (645) 39 4415 195 (653)U100020 222 222 22219 113 (347) 222 22207 151 (327)U100040 737 737 73700 120 (243) 737 73700 177 (205)1Best result from the literature2Best and average results from 1000 runs3CPU seconds on a Pentium IV 28GHz4Average number of generations over 1000 runs

Table 5 Comparison of GEFM and CMPA on 8-way partitioning

Graph GEFM [21] CMPABest1 Ave1 CPU2 (Gen3) Best1 Ave1 CPU2 (Gen3)

G50025 111 11521 4203 (2090) 111 11451 4025 (2091)G50005 465 46816 8565 (2314) 465 46763 7296 (2239)G50010 1254 125953 14454 (2347) 1254 125847 13504 (2297)G50020 3348 335480 29318 (2437) 3348 335371 28509 (2317)G100025 212 21630 14676 (3043) 211 21657 14842 (3023)G100005 930 93811 24720 (3148) 931 93926 23467 (3069)G100010 2714 272633 40896 (3146) 2711 272552 41698 (3083)G100020 6525 653655 79129 (3303) 6520 653835 78167 (3141)U50005 16 1880 7087 (2629) 16 1722 5589 (2137)U50010 143 14512 9875 (2277) 143 14426 9050 (1984)U50020 612 61457 12122 (1690) 611 61293 9753 (1483)U50040 1867 187260 15794 (1396) 1867 187173 12962 (1204)U100005 21 3366 18035 (3638) 20 2853 20953 (3172)U100010 176 18370 24308 (3105) 176 18230 25702 (2857)U100020 812 81405 23108 (1824) 812 81300 23293 (1756)U100040 2562 256305 24946 (1353) 2562 256285 27351 (1238)1Best and average results from 100 runs2CPU seconds on a Pentium IV 28GHz3Average number of generations over 100 runs

heuristic but KL-MA andGEFM do notWe performed 1000runs for bipartitioning and 100 runs for 8-way and 32-waypartitioning All the programs were written in the C languageand compiled using GNUrsquos gcc compiler It was run on aPentium IV 28GHz computer with Linux In the tables thebold-faced numbers indicate the lower of the average cut sizes

obtained by the two algorithms CMPA outperformed KL-MA and GEFM on most graphs with comparable runningtimes

CMPA underperformed on some random graphs whichmay be due to the weak cluster structures in these graphsCMPArsquos average performance was better on all the geometric

8 Mathematical Problems in Engineering

Table 6 Comparison of GEFM and CMPA on 32-way partitioning

Graph Best known1 GEFM [21] CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

G50025 177 177 18169 7898 (1491) 177 18082 7430 (1382)G50005 623 624 63047 16226 (2197) 623 62992 15918 (2035)G50010 1573 1574 158194 30083 (2364) 1572lowast 157972 35573 (2332)G50020 4034 4037 404506 68153 (2565) 4035 404281 67319 (2479)G100025 312 313 32099 33579 (2798) 313 32025 35644 (2648)G100005 1217 1208 121872 69867 (3133) 1205lowast 121689 73806 (3203)G100010 3360 3353lowast 336731 117148 (3294) 3355 336608 132359 (3267)G100020 7818 7817 782981 201278 (3475) 7815lowast 783048 236369 (3342)U50005 109 112 11639 13829 (1640) 109 11318 12115 (1595)U50010 523 531 53775 22552 (1512) 526 53184 16626 (1222)U50020 1825 1831 184139 28507 (1344) 1823lowast 183188 29947 (1197)U50040 5328 5364 538001 56174 (1381) 5348 536983 52315 (1331)U100005 117 118 12602 45103 (3137) 115lowast 12349 46435 (2967)U100010 577 576 58326 59901 (2599) 571lowast 57816 74350 (2603)U100020 2367 2375 239633 79877 (1963) 2373 238893 93043 (1848)U100040 7329 7399 741749 142178 (1634) 7382 740718 149310 (1573)1Best known values [20ndash22 24 33]2Best and average results from 100 runs Asterisked numbers are new best values3CPU seconds on a Pentium IV 28GHz4Average number of generations from 100 runs

graphs This suggests that effective cluster handling is moreimportant on the geometric graphs as we suggested inSection 31

The local optimization algorithm ismuchmore expensivethan the cluster-handling heuristic thus CMPA does nottake much longer to run than KL-MA or GEFM On averageCMPA required 1414more time than KL-MA for the bipar-titioning problems and 202 more than GEFM in 32-waypartitioning CMPA ran 552 faster in 8-way partitioning

7 Concluding Remarks

We devised a cluster-moving heuristic and combined it witha memetic algorithm (MA) for graph partitioning Experi-ments on 2-way 8-way and 32-way partitioning showed thatthis heuristic significantly improved the performance of theMA especially on the 32-way partitioning

The method of moving clusters that we have introducedaddresses a significant weakness in standard IIP algorithmsThe idea of adding an operation to complement a localoptimization algorithm might be used to address otherdeficiencies in MAs

Our method of cluster detection is based on a measurethat we call genic distance which is designed to reflect theextent to which two vertices connected by an edge belongto the same cluster Instead of computing genic distancesonce in an initialization phase anMA could recompute thesedistances as it progresses this might improve the accuracyof cluster detection at some cost in time We believe thatgenic distance might also be useful in solving other problemsinvolving clusters such as the maximum clique problem

Table 7 Real-world benchmark graphs

Graph Number of vertices Number of edgesnasa4704 4704 50026bcspwr09 1723 2394bcsstk13 2003 40940DEBR12 4096 8189

Appendix

Results on Real-World Graphs

We also tested on four real-world benchmark graphs usedin [11 15 43] The sizes of the graphs are given in Table 7We conducted experiments on 2-way and 8-way partitioningWe performed 100 runs for bipartitioning and 50 runs for8-way partitioning It was run on an Intel Core i7 36GHzcomputer with Linux Table 8 compares CMPA with KL-MAon bipartitioning and Table 9 compares CMPA with GEFMon 8-way partitioning In the tables the bold-faced numbersindicate the lower of the average cut sizes obtained by the twoalgorithms Similarly to the results in Section 6CMPAoverallperformed better than the others with comparable runningtimes

Disclosure

Apreliminary version of this paper appeared inProceedings ofthe Genetic and Evolutionary Computation Conference 2007(p 1520)

Mathematical Problems in Engineering 9

Table 8 Comparison of KL-MA and CMPA on bipartitioning

Graph Best known1 KL-MA CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

nasa4704 1292 1292 129200 415 (407) 1292 129200 382 (347)bcspwr09 9 9 1131 061 (1010) 9 1060 081 (1003)bcsstk13 2355 2355 235500 117 (330) 2355 235500 146 (282)DEBR12 548 548 54844 991 (765) 548 54820 1032 (794)1Best result from the literature2Best and average results from 100 runs3CPU seconds on Intel Core i7 36GHz4Average number of generations over 100 runs

Table 9 Comparison of GEFM and CMPA on 8-way partitioning

Graph GEFM [21] CMPABest1 Ave1 CPU2 (Gen3) Best1 Ave1 CPU2 (Gen3)

nasa4704 3898 390366 57569 (2235) 3896 390266 59151 (2255)bcspwr09 53 5708 7880 (2947) 54 5774 8516 (2925)bcsstk13 8911 893910 28119 (1427) 8919 893614 30154 (1476)DEBR12 1248 126016 31597 (3969) 1248 125990 35383 (4002)1Best and average results from 50 runs2CPU seconds on Intel Core i7 36GHz3Average number of generations over 50 runs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the ICT RampD program ofMSIPIITP Korea (10045253 The development of SDNOpenFlow-Based Enterprise Network Controller Technol-ogy) project The authors would like to thank ProfessorByung-Ro Moon and Dr Yongjoo Song for their valuablesuggestions for improving this paper The authors also thankJong-Pil Kim for providing the source code of GEFM [21]

References

[1] Y Kanizo D Hay and I Keslassy ldquoPalette distributing tablesin software-defined networksrdquo in Proceedings of the IEEEInternational Conference on Computer Communications pp545ndash549 April 2013

[2] P Wette M Draxler and A Schwabe ldquoMaxiNet distributedemulation of software-defined networksrdquo in Proceedings of theIFIP Networking Conference pp 1ndash9 June 2014

[3] M R Garey and D S Johnson Computers and Intractability AGuide to the Theory of NP-Completeness W H Freeman 1979

[4] T N Bui and C Jones ldquoFinding good approximate vertex andedge partitions is NP-hardrdquo Information Processing Letters vol42 no 3 pp 153ndash159 1992

[5] Y-H Kim ldquoImproved implementation choices for iterativeimprovement partitioning algorithms on circuitsrdquo in Proceed-ings of the International Conference on Computer Design pp 30ndash34 July 2008

[6] Y Yoon and Y-H Kim ldquoNew bucket managements in iterativeimprovement partitioning algorithmsrdquo Applied Mathematics ampInformation Sciences vol 7 no 2 pp 529ndash532 2013

[7] BW Kernighan and S Lin ldquoAn efficient heuristic procedure forpartitioning graphsrdquo Bell System Technical Journal vol 49 no2 pp 291ndash307 1970

[8] C Fiduccia and R Mattheyses ldquoA Linear-time heuristic forimproving network partitionsrdquo in Proceedings of the Conferenceon Design Automation pp 175ndash181 June 1982

[9] D S Johnson C R Aragon L A McGeoch and C SchevonldquoOptimization by simulated annealing An experimental evalu-ation Part I Graph partitioningrdquo Operations Research vol 37no 6 pp 865ndash892 1989

[10] T N Bui and B R Moon ldquoGenetic algorithm and graphpartitioningrdquo IEEETransactions on Computers vol 45 no 7 pp841ndash855 1996

[11] R Battiti and A A Bertossi ldquoGreedy prohibition and reactiveheuristics for graph partitioningrdquo IEEE Transactions on Com-puters vol 48 no 4 pp 361ndash385 1999

[12] E Rolland H Pirkul and F Glover ldquoTabu search for graphpartitioningrdquo Annals of Operations Research vol 63 no 2 pp209ndash232 1996

[13] T N Bui and L C Strite ldquoAn ant system algorithm for graphbisectionrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference pp 43ndash51 July 2002

[14] I Hwang Y-H Kim and B-R Moon ldquoMulti-attractor genereordering for graph bisectionrdquo inProceedings of theGenetic andEvolutionary Computation Conference pp 1209ndash1215 July 2006

[15] Y-H Kim and B-R Moon ldquoLock gain based graph partition-ingrdquo Journal of Heuristics vol 10 no 1 pp 37ndash57 2004

[16] P Merz and B Freisleben ldquoFitness landscapes memetic algo-rithms and greedy operators for graph bipartitioningrdquo Evolu-tionary Computation vol 8 no 1 pp 61ndash91 2000

[17] HMuhlenbein and TMahnig ldquoEvolutionary optimization andthe estimation of search distributions with applications to graph

10 Mathematical Problems in Engineering

bipartitioningrdquo International Journal of Approximate Reasoningvol 31 no 3 pp 157ndash192 2002

[18] A G Steenbeek E Marchiori and A E Eiben ldquoFindingbalanced graph bi-partitions using a hybrid genetic algorithmrdquoin Proceedings of the IEEE International Conference on Evolu-tionary Computation pp 90ndash95 May 1998

[19] Y Yoon and Y-H Kim ldquoVertex ordering clustering andtheir application to graph partitioningrdquoApplied Mathematics ampInformation Sciences vol 8 no 1 pp 135ndash138 2014

[20] S-J Kang and B-R Moon ldquoA hybrid genetic algorithm formultiway graph partitioningrdquo in Proceedings of the Genetic andEvolutionary Computation Conference pp 159ndash166 July 2000

[21] J-P Kim and B-R Moon ldquoA hybrid genetic search for multi-way graph partitioning based on direct partitioningrdquo in Pro-ceedings of the Genetic and Evolutionary Computation Confer-ence pp 408ndash415 July 2001

[22] Y-H Kim A Moraglio Y Yoon and B-R Moon ldquoGeometriccrossover for multiway graph partitioningrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1217ndash1224 July 2006

[23] A Moraglio Y-H Kim Y Yoon and B-R Moon ldquoGeometriccrossovers for multiway graph partitioningrdquo Evolutionary Com-putation vol 15 no 4 pp 445ndash474 2007

[24] AMoraglio Y-H Kim Y Yoon B-RMoon and R Poli ldquoGeo-metric crossover for permutations with repetitions applicationto graph partitioningrdquo in Proceedings of the PPSN Workshopon Evolutionary AlgorithmsmdashBridging Theory and PracticeSeptember 2006

[25] A J Soper C Walshaw and M Cross ldquoA combined evolu-tionary search and multilevel optimisation approach to graph-partitioningrdquo Journal of Global Optimization vol 29 no 2 pp225ndash241 2004

[26] Y Yoon Y-H Kim A Moraglio and B-R Moon ldquoQuotientgeometric crossovers and redundant encodingsrdquo TheoreticalComputer Science vol 425 pp 4ndash16 2012

[27] J Kim I Hwang Y-H Kim and B-R Moon ldquoGeneticapproaches for graph partitioning a surveyrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 473ndash480 July 2011

[28] S Dutt and W Deng ldquoA probability-based approach to VLSIcircuit partitioningrdquo in Proceedings of the Design AutomationConference pp 100ndash105 June 1996

[29] B Krishnamurthy ldquoAn improved min-cut algorithm for parti-tioning VLSI networksrdquo IEEE Transactions on Computers vol33 no 5 pp 438ndash446 1984

[30] L A Sanchis ldquoMultiple-way network partitioningrdquo IEEETrans-actions on Computers vol 38 no 1 pp 62ndash81 1989

[31] G Laszewski ldquoIntelligent structural operators for the k-waygraph partitioning problemrdquo in Proceedings of the InternationalConference on Genetic Algorithms pp 45ndash52 July 1991

[32] H Muhlenbein ldquoParallel genetic algorithms in combinatorialoptimizationrdquo in Computer Science and Operations Research OBalci R Sharda and S Zenios Eds pp 441ndash453 PergamonOxford UK 1992

[33] S-S Choi and B-R Moon ldquoNormalization in genetic algo-rithmsrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference vol 2723 of Lecture Notes in ComputerScience pp 862ndash873 2003

[34] S-H Kim Y-H Kim and B-R Moon ldquoA hybrid geneticalgorithm for themax cut problemrdquo inProceedings of theGeneticand Evolutionary Computation Conference pp 416ndash423 July2001

[35] Y-H Kim Y-K Kwon and B-R Moon ldquoProblem-independ-ent schema synthesis for genetic algorithmsrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1112ndash1122 July 2003

[36] K Seo S Hyun and Y-H Kim ldquoAn edge-set representationbased on spanning tree for searching cut spacerdquo IEEE Trans-actions on Evolutionary Computation 2014

[37] Y-H Kim Combinatorial optimization based on effective localsearch genetic algorithms and problem space analyses [PhDthesis] Seoul National University Seoul Republic of Korea2005

[38] Y-H Kim ldquoAn enzyme-inspired approach to surmount barriersin graph bisectionrdquo in Proceedings of the International Confer-ence on Computational Science and Its Applications vol 5072 ofLecture Notes in Computer Science pp 841ndash851 2008

[39] S Dutt and W Deng ldquoVLSI circuit partitioning by cluster-removal using iterative improvement techniquesrdquo in Proceed-ings of the IEEEACM International Conference on Computer-Aided Design pp 194ndash200 November 1996

[40] S Dutt and W Deng ldquoCluster-aware iterative improvementtechniques for partitioning large VLSI circuitsrdquo ACM Transac-tions on Design Automation of Electronic Systems vol 7 no 1pp 91ndash121 2002

[41] Y Yoon and Y-H Kim ldquoAn efficient genetic algorithm formaximum coverage deployment in wireless sensor networksrdquoIEEE Transactions on Cybernetics vol 43 no 5 pp 1473ndash14832013

[42] Y Yoon Y-H Kim AMoraglio and B-RMoon ldquoA theoreticaland empirical study on unbiased boundary-extended crossoverfor real-valued representationrdquo Information Sciences vol 183no 1 pp 48ndash65 2012

[43] BMonien andRDiekmann ldquoA local graph partitioning heuris-tic meeting bisection boundsrdquo in Proceedings of the SIAM Con-ference on Parallel Processing for Scientific Computing March1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

by clusters by modifying the GA so that it contributes tomove clusters appropriatelyWe present a memetic algorithm(MA) which is a GA combined with local optimizationin which a heuristic finds clusters in some of the positionsin each generation by examining population of individualseach of which represents a position rather than trying toidentify them directly from a single graph It moves someof these clusters This heuristic supplements the well-knownability of MAs to provide attractive initial points for localoptimization Experimental results show that this approachcan substantially improve the performance of an MA Thecontributions of this work are summarized as follows

(i) We provide a detailed explanation of the difficultyof moving clusters in graph partitioning and provideexperimental results quantifying the impact of clus-ters on the search for partitions with a small cut size

(ii) We present a heuristic for detecting and movingclusters which is based on a new population-basedmeasure of the distance between vertices called genicdistance

(iii) We show that this heuristic substantially improves theability of an MA to find good partitions

The remainder of this paper is organized as follows InSection 2 we briefly introduce IIP algorithms and the testgraphs used in our experiments In Section 3 we investigatethe difficulty of moving clusters in graph partitioning InSection 4we describe our new cluster-handling heuristic andan MA that uses this heuristic is described in Section 5 InSection 6 we present experimental results and draw conclu-sions in Section 7

2 Preliminaries

21 Iterative Improvement Algorithms in Bipartitioning Itera-tive improvement partitioning starts with a randompartitionThis is refined in a series of passes At the start of each passall the vertices are free to move between subsets IIP selectsvertices and moves them but each vertex is only moved onceduring a pass At the end of the pass the best partition foundduring the pass is identified and used as the input to the nextpass Passes continue until there is no further improvement

There are a number of IIP algorithms of which KL [7]is often considered to be the first reasonable heuristic forbipartitioning In KL the movement of vertices during a passis restricted to the swapping of a pair of vertices betweensubsets

Let 119860 119861 be a partition of 119881 into two subsets 119860 and 119861We define the gain 119892V associated with a vertex V to be thereduction in cut size obtained bymoving V to the other subsetBy extension the gain 119892(119886 119887) obtained by swapping vertices119886 isin 119860 and 119887 isin 119861 can be expressed as follows

119892 (119886 119887) = 119892119886+ 119892119887minus 2120575 (119886 119887) (1)

where

120575 (119886 119887) = 1 if (119886 119887) isin 1198640 otherwise

(2)

KL selects the pair (119886 119887) with the highest value of 119892(119886 119887)and effects the exchange The vertices 119886 and 119887 are notconsidered again during the current pass A sequence of pairs(1198861 1198871) (1198862 1198872) (119886

1198992minus1 1198871198992minus1

) are selected in this wayThe algorithm chooses 119897 that maximizes sum119897

119894=1119892(119886119894 119887119894) and

exchanges 1198861 1198862 119886

119897 and 119887

1 1198872 119887

119897 KL performs

further passes until no improvement is possibleFM is another widely used IIP algorithm which is similar

to KL except that it only moves one vertex at a time Thismakes FM faster than KL with little loss in partition qualitySeveral variants of KL and FM exist [15 28 29]

22 Local Optimization Algorithms for Multiway PartitioningThere are three main schemes for multiway partitioningwhich are developments of the recursive pair-wise and directapproaches [21] to bipartitioningThe recursive KL algorithmbisects the graph recursively until there are 119896 subsets Thepair-wise KL [7] starts with an arbitrary 119896-way partition Itpicks two subsets at a time from the 119896 subsets and performsbipartitioning to reduce the cut size between those pairs San-chis [30] extended the FMalgorithm tomultiway partitioningand showed that the direct method performed better thanrecursion The extended algorithm considers moving eachvertex from its current subset to every other subset Toperform local optimization in the proposedMA formultiwaypartitioning we use a variant of this algorithm called EFM(extended FM) [21] The time complexity of EFM is 119874(119896|119864|)

23 Local Search in Memetic Algorithms It is already clearthat combining a GA with local optimization algorithmsis an effective approach to the graph partitioning problem[15] Some authors have explored fast but weak local opti-mization algorithms For example [31 32] 2-opt was usedto relocate border vertices which are those with edges thatconnect to vertices in other subsets Bui and Moon [10]obtained better results with KL by allowing only a single passwhile restricting the number of vertices to be swappedConversely other authors have reported notable improve-ments by enhancing local optimization algorithms For bipar-titioning Kim and Moon [15] suggested a new KL-basedlocal optimization algorithm formulated using a new typeof gain called lock gain which only takes into account theedges that connect a vertex to the vertices that have alreadybeen moved Combined with a GA this algorithm obtainedimpressive results on most benchmark graphs For multi-way partitioning the combination of MAs with specializedlocal optimization algorithms showed good results [20 21]Steenbeek et al [18] proposed what they called a clusterenhancement heuristic which they combined with an MAand reported successful results Their MA uses a vertexswap heuristic to identify clusters The MA only handles themoving of clusters between subsets

24 Test Graphs We tested our MA on Johnsonrsquos benchmarkgraphs [9] which have been widely used in other studies [1011 14ndash17 20 21 23 33ndash36] They are composed of 8 random

Mathematical Problems in Engineering 3

Cluster

3

4

1

2

5 edges connecting subsets

(a) Before moving a cluster

Cluster

3

4

1

2

1 edge connecting subsets

(b) After moving a cluster

Figure 1 An example of cluster moving in which the cut size of apartition is reduced by 4

graphs G119899119889 and 8 random geometric graphs U119899119889 The twodifferent classes of graphs are briefly described below

(i) G119899119889 a random graph on 119899 vertices with an indepen-dent probability 119901 that any two vertices are connectedby an edge The probability 119901 is biased so that theexpected vertex degree 119901(119899 minus 1) is 119889

(ii) U119899119889 a random geometric graph on 119899 vertices that liein the unit square and whose coordinates are chosenuniformly from the unit interval Every pair of ver-tices separated by a distance of 119905 or less is connectedby an edge The expected degree of a vertex is 1198991205871199052

3 Difficulty of Moving Clusters

Suppose that the cluster shown in Figure 1(a) is involved ina bipartitioning problem The four vertices in this cluster arefully interconnected and they all belong to the same subsetMoving this cluster to the other subset across the dottedline in Figure 1(b) will reduce the cut size of the partition by4 However there is no motivation to move any single vertexbecause they all have negative gain the gains of V

1 V2 V3 and

V4are minus1 minus4 minus2 and minus1 respectivelyThis example illustrates

how IIP algorithms may miss a significant reduction incut size that could be achieved by moving several verticestogether

The baleful effect of clusters on local search algorithmstrying to solve the graph partitioning problemmotivated thisstudy Kim [37 38] indicated that graph partitioning is hardprimarily due to the difficulty of moving clusters Dutt andDeng [39 40] have also observed that an IIP method appliedto circuit partitioning can fail because of the difficulty ofdealing with clusters that straddle subsets

31 Experimental Support We designed experiments toquantify the effect of clusters on IIP algorithms representedby the KL algorithm Using the cluster detection method tobe described in Section 41 we find clusters in the graph andselect one randomly We then take a locally optimum bipar-tition 119904 obtained by KL and move the selected cluster to theother subset creating a perturbed partition 119905cluster ApplyingKL to 119905cluster we obtain a new local optimum 119906cluster

Table 1 Probability that KL fails to return vertices moved from onesubset of a partition to the other when the vertices are in cluster(119875cluster) or chosen at random (119875random) over 1000 runs

Graph 119875cluster () 119875random () Average number of verticesmoved (119902)

G50025 1690 250 641

G50005 570 130 550

G50010 190 030 505

G50020 050 010 478

G100025 1280 190 589

G100005 290 080 561

G100010 060 020 467

G100020 030 000 491

U50005 3910 100 741

U50010 2620 050 1205

U50020 860 020 1410

U50040 1040 040 4936

U100005 3790 080 781

U100010 2850 060 1247

U100020 1160 010 2219

U100040 910 040 3588

Assuming that 119902 the number of vertices in the clusters issmall 119906cluster can be expected to match 119904 if KL is successfulin moving the cluster back However if KL fails to returnthe perturbed cluster to its original subset the cut size of thepartition may increase Repeating this experiment we derive119875cluster as an approximation of the probability that the cut sizeof 119906cluster is larger than that of 119904

For comparison we perturbed 119902 vertices randomlyselected within a locally optimized partition bymoving themto the other subset We call this partition 119905random We applythe KL algorithm to 119905random and then obtain a new locallyoptimum partition 119906random Repeating this experiment wederive 119875random as an approximation of the probability that thecut size of 119906random is larger than that of 119904

Table 1 shows the values of 119875cluster and 119875random for 16benchmark graphs We see that 119875cluster is always larger than119875random as we would expect We notice that the gap between119875cluster and 119875random is larger on the geometric graphs (U119899119889)than on the random graphs (G119899119889)

4 Cluster-Handling Heuristic

Cluster analysis is a well-known problem for which plenty ofalgorithms exist many of which require a lot of computationThe insight that motivates our heuristic is that the applicationof a local optimization process such as IIP to a randomly par-titioned graph creates a modified partition in which clusterstend to be wholly allocated to one subset or another (and arethen difficult to move as we have already observed) A singlepartition of this sort is of little help in identifying clustersbecause the clusters are not separated at all within eachsubset but if we create many random partitions and optimize

4 Mathematical Problems in Engineering

Table 2 Truth table for Fact 1

119901 119902 119901 rarr 119902 119868(119901) 119868(119902)

True True True 1 1

True False False 1 0

False True True 0 1

False False True 0 0

them we can reasonably infer that vertices that find them-selves in the same subset in most of these partitions belong tothe same cluster We can make this inference in a structuredway using the ldquogenic distancerdquo metric that we will introduceThis approach to cluster analysis may seem indirect but it isefficient in the context of an evolutionary approach to graphpartitioning because the set of partitions required for findingclusters using genic distance is also the population which wemust create to be evolved by our MA

One way of dealing with clusters is to devise a local opti-mization heuristic that can identify clusters [18 19 38] How-ever this prevents us from building on previous studies of IIPalgorithms

Our approach is to add an additional heuristic to ourMA which finds and moves clusters The heuristic identifiesclusters in the population of partitions which have alreadybeen optimized locally It selects clusters with higher gainsandmoves them IIP local optimization is then applied again

41 Cluster Detection Let 119868(sdot) be an indicator function that is119868(true) = 1 and 119868(false) = 0 Then we can trivially establishthe following

Fact 1 If 119901 rarr 119902 is true then 119868(119901) le 119868(119902)

Proof From Table 2

Fact 2 119868(119901 or 119902) le 119868(119901) + 119868(119902)

Proof From Table 3

We now define a metric called genic distance which mea-sure the extent to which two vertices connected by an edgecan be considered to belong to the same cluster We denotethe genic distance of an edge V

119894 V119895 within a population

1199011 1199012 119901

119898 of locally optimized partitions as 119889

119892(V119894 V119895)

which can be expressed as follows

119889119892(V119894 V119895) =

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119895)) (3)

where 119892119894and 119892

119895are the genes corresponding to V

119894and V

119895

respectively The value of gene 119892119894(ie the partition number

that vertex V119894belongs to) in the 119897th individual is 119901

119897(119892119894) For

convenience we assume that each vertex has an edge thatconnects it to itself so that V V isin 119864 for each vertex V Thenthe following proposition holds

Proposition 1 For each population 119889119892becomes a pseudomet-

ric on 119881

Table 3 Truth table for Fact 2

119901 119902 119868(119901) 119868(119902) 119901 or 119902 119868(119901 or 119902) 119868(119901) + 119868(119902)True True 1 1 True 1 2

True False 1 0 True 1 1

False True 0 1 True 1 1

False False 0 0 False 0 0

Proof Since 119868(sdot) ge 0 119889119892(V119894 V119895) = sum

119898

119897=1119868(119901119897(119892119894) = 119901119897(119892119895)) ge 0

for each V119894 V119895 isin 119864 It is enough to show the following three

conditions

(i) 119889119892(V119894 V119894) = 0

119889119892(V119894 V119894) =

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119894))

=

119898

sum

119897=1

119868 (false)

= 0

(4)

(ii) Symmetry Let V119894 V119895 be in 119864

119889119892(V119894 V119895) =

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119895))

=

119898

sum

119897=1

119868 (119901119897(119892119895) = 119901119897(119892119894))

= 119889119892(V119895 V119894)

(5)

(iii) Triangle Inequality Consider each group of threeedges V

119894 V119895 V119894 V119896 V119896 V119895 isin 119864 If 119901

119897(119892119894) = 119901

119897(119892119896)

and119901119897(119892119896) = 119901119897(119892119895) then119901

119897(119892119894) = 119901119897(119892119895) for each 119897 By

contraposition if 119901119897(119892119894) = 119901119897(119892119895) then 119901

119897(119892119894) = 119901119897(119892119896)

or 119901119897(119892119896) = 119901119897(119892119895)

For each 119897

119868 (119901119897(119892119894) = 119901119897(119892119895))

le 119868 (119901119897(119892119894) = 119901119897(119892119896) or 119901119897(119892119896) = 119901119897(119892119895)) (∵ Fact 1)

le 119868 (119901119897(119892119894) = 119901119897(119892119896)) + 119868 (119901

119897(119892119896) = 119901119897(119892119895)) (∵ Fact 2)

(6)

By summing the above inequalities for all 119897

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119895))

le

119898

sum

119897=1

(119868 (119901119897(119892119894) = 119901119897(119892119896)) + 119868 (119901

119897(119892119896) = 119901119897(119892119895)))

=

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119896)) +

119898

sum

119897=1

119868 (119901119897(119892119896) = 119901119897(119892119895))

(7)

Mathematical Problems in Engineering 5

Therefore we have

119889119892(V119894 V119895) le 119889119892(V119894 V119896) + 119889119892(V119896 V119895) (8)

That is 119889119892satisfies the triangle inequality

Proposition 1 suggests that the measure 119889119892is reasonable

A pseudometric space is a generalization of a metric space inwhich points need not be distinguishable thus it is possiblethat 119889

119892(V119894 V119895) = 0 for some edge V

119894 V119895 with distinct vertices

V119894= V119895

Our heuristic detects clusters by collecting a number oflocal optima and computes genic distances for all the edges ineach graphThis takes119874(119898|119864|) time but this cost is negligiblesince this computation is a preprocess performed before theMA runs The heuristic temporarily eliminates edges withgenic distances that are greater than a threshold value 120579 Weset 120579 to be the smallest value that satisfies

10038161003816100381610038161003816119890 isin 119864 119889

119892(119890) le 120579

10038161003816100381610038161003816le 01 |119881| (9)

Each remaining connected component containingmore thanthree vertices is considered to be a cluster

Figure 2 shows how our heuristic detects clusters Fig-ure 2(a) shows an example graph with 11 vertices and 15edges Four individuals corresponding to locally optimizedpartitions from the population are shown in Figure 2(b)In Figure 2(c) each edge is labeled with its genic distanceIf the threshold value of genic distance is 1 then theedges with larger genic distances indicated by dotted lineare eliminated Then four connected components remainV3 V7 V8 V9 V2 V4 V6 V11 V1 V5 and V

10 The last two

of these connected components are considered too small tobe clusters Thus clusters V

3 V7 V8 V9 and V

2 V4 V6 V11

shaded in Figure 2(c) remain as candidates for moving

42 Cluster-Moving Scheme Our heuristic improves theoffspring of each generation after crossover by moving theclusters that were detected using the technique described inthe previous subsection To select the clusters to be movedand their target subsets we introduce ameasure called clustergain such that cg(119909 119886) is the reduction in the cut size of thepartition when all the vertices in cluster 119909 are moved to thesubset 119886 For example moving the cluster in Figure 1(a) to theother subset in the partition is associated with a cluster gainof 2

This cluster-moving scheme described in Algorithm 1is applied to each individual generated by crossover whichis a partition that may be unbalanced However cut sizeand cluster gain are well defined on unbalanced partitionsOur scheme does not consider moving every cluster in everypartition because we found that making all clusters movablecauses the premature convergence of the MA Thus ourheuristic selects119873 clusters at random as candidates for mov-ing In our experiments we set119873 to 5We compute the clustergain that results from moving each candidate cluster to eachof the other 119896 minus 1 subsets The candidate cluster 119909 anddestination subset 119886with the highest cluster gain are selectedAssume that cg(119909 119886) is positive all the vertices in cluster 119909

6

12

3 45

78

9 1011

(a) Example graph

1

1 1 1 1 1 1

1 113 5 7 8 11

1

1111110 0 0 0 0

10 0

0 0 0 0 0

1 1 1 1 10 00 0 0 0

0 0 0 010642 9

(b) Four locally optimized partitionsin the population (subsets 0 and 1)

00

00

0

00

11

1

23

03

3

(c) Two detected clusters

Figure 2 An example of cluster detection

are moved to subset 119886 and cluster 119909 is removed from theset of candidate clusters This process is repeated until nocandidates remain or no move yields a positive cluster gainNo attempt is made to balance the partition during cluster-moving this is performed later

5 Memetic Search Framework

CMPA (cluster-moving memetic partitioning algorithm) is amemetic algorithm that we have designed for graph parti-tioning In this MA an individual is a 119896-way partition Eachgene in an individual corresponds to a vertex and has a valuebetween 0 and 119896 minus 1 which indicates the subset to which thevertex belongs that is the 119894th gene 119892

119894= 119895 hArr V

119894isin 119875119895+1

It isa steady-state MA meaning that there is only one offspringfrom population in each generation Crossover is followed bya cluster-moving step and then local optimization

Algorithm 2 shows the processes that make up CMPAwhich we now describe in detail

(i) Initialization When the MA starts 119898 individuals(ie partitions) are created at random Then eachindividual is improved by local optimization We set119898 to be 30 for bipartitioning and 50 for multiwaypartitioning (with 119896 = 8 or 119896 = 32)

6 Mathematical Problems in Engineering

Select clusters 119878 larr 119873 clusters selected at randomdo

Calculate cg(119909 119886) for all 119909 isin 119878 1 le 119886 le 119896cglowast larr max

119909119886cg(119909 119886)

(119909 119886) larr argmax119909119886

cg(119909 119886)

if cglowast gt 0 then

Move cluster 119909 to partition 119886119878 larr 119878 119909

until (119878 = 0 or cglowast le 0)

Algorithm 1 Our cluster-moving scheme

Create initial population of fixed sizeApply local optimization to each member of populationCalculate genic distance from populationFind clusters and store their informationdo

Select 1199011198861199031198901198991199051and 119901119886119903119890119899119905

2from population

Normalization(1199011198861199031198901198991199051 119901119886119903119890119899119905

2)

offspring larr Crossover(1199011198861199031198901198991199051 119901119886119903119890119899119905

2)

Cluster-moving(offspring)Local-optimization(offspring)Replace(population offspring)

until (stopping condition)return the best solution

Algorithm 2 The process in CMPA

(ii) Selection We used the roulette-wheel-based propor-tional selection The probability that the best individ-ual is chosen was set to four times the probabilitythat the worst is chosen The fitness value 119891

119894of the

119894th individual is expressed as (119888119908minus 119888119894) + (119888119908minus 119888119887)3

where 119888119887 119888119908 and 119888

119894are the cut sizes of the partitions

corresponding to the best the worst and the 119894thindividual respectively

(iii) Normalization Laszewski [31] first used normaliza-tion to improve the performance of GA and its vari-ants have been suggested in [23 26 41 42]The parentindividuals are normalized before crossover followingLaszewski [31 33] The subset of one parent whichshares the largest number of vertices with subsetsof the other parent is selected and that subset isnumbered 0 This process is repeated incrementingthe index until all subsets are numbered

(iv) Crossover and Cluster Moving We used a standardfive-point crossover After crossover the cluster-han-dling heuristic described in Section 42 is applied tothe individual At this point individuals usually cor-respond to unbalanced partitionsWe select a randomlocation in the individual and adjust the values ofthe genes which are the subsets to which the corre-sponding vertices belong to the right of this location(in a typical circular string) until the partition is

balanced This is effectively a mutation effect and nofurther mutation was introduced

(v) Local Optimization KL [7] was used for the bisectionproblems and EFM (extended FM) [21] was used formultiway partitioning (119896 = 8 or 119896 = 32)

(vi) Replacement We used a replacement scheme due toBui and Moon [10] If an offspring is better than itscloser parent the MA replaces that parent If it is bet-ter than its other parent then that parent is replacedOtherwise it replaces the worst individual in thepopulation

(vii) Stopping Condition This is based on consecutive fail-ures to replace an individualrsquos parents Termination istriggered by consecutive failures 30 in bipartitioningand 50 in multiway partitioning

6 Experimental Results

We conducted experiments on 2-way 8-way and 32-way par-titioning Table 4 shows the performance of MA combinedwith KL (denoted KL-MA) and CMPA on bipartitioningTable 5 shows the performance of the genetic extended FMalgorithm (GEFM) [21] one of themost effective approachesand CMPA on 8-way partitioning and Table 6 gives theresults for 32-way partitioning CMPAuses a cluster-handling

Mathematical Problems in Engineering 7

Table 4 Comparison of KL-MA and CMPA on bipartitioning

Graph Best known1 KL-MA CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

G50025 49 49 5134 032 (595) 49 5122 034 (558)G50005 218 218 21865 059 (700) 218 21845 064 (682)G50010 626 626 62755 102 (681) 626 62746 110 (672)G50020 1744 1744 174583 193 (694) 1744 174562 209 (687)G100025 93 93 9726 096 (745) 93 9708 104 (722)G100005 445 445 45150 207 (884) 445 45092 222 (860)G100010 1362 1362 136668 407 (969) 1362 136618 445 (977)G100020 3382 3382 338561 742 (971) 3382 338526 813 (987)U50005 2 2 496 055 (793) 2 394 066 (803)U50010 26 26 2631 045 (459) 26 2604 057 (465)U50020 178 178 17806 042 (300) 178 17800 053 (271)U50040 412 412 41200 034 (190) 412 41200 037 (102)U100005 1 1 1167 171 (1077) 1 881 212 (1068)U100010 39 39 4627 151 (645) 39 4415 195 (653)U100020 222 222 22219 113 (347) 222 22207 151 (327)U100040 737 737 73700 120 (243) 737 73700 177 (205)1Best result from the literature2Best and average results from 1000 runs3CPU seconds on a Pentium IV 28GHz4Average number of generations over 1000 runs

Table 5 Comparison of GEFM and CMPA on 8-way partitioning

Graph GEFM [21] CMPABest1 Ave1 CPU2 (Gen3) Best1 Ave1 CPU2 (Gen3)

G50025 111 11521 4203 (2090) 111 11451 4025 (2091)G50005 465 46816 8565 (2314) 465 46763 7296 (2239)G50010 1254 125953 14454 (2347) 1254 125847 13504 (2297)G50020 3348 335480 29318 (2437) 3348 335371 28509 (2317)G100025 212 21630 14676 (3043) 211 21657 14842 (3023)G100005 930 93811 24720 (3148) 931 93926 23467 (3069)G100010 2714 272633 40896 (3146) 2711 272552 41698 (3083)G100020 6525 653655 79129 (3303) 6520 653835 78167 (3141)U50005 16 1880 7087 (2629) 16 1722 5589 (2137)U50010 143 14512 9875 (2277) 143 14426 9050 (1984)U50020 612 61457 12122 (1690) 611 61293 9753 (1483)U50040 1867 187260 15794 (1396) 1867 187173 12962 (1204)U100005 21 3366 18035 (3638) 20 2853 20953 (3172)U100010 176 18370 24308 (3105) 176 18230 25702 (2857)U100020 812 81405 23108 (1824) 812 81300 23293 (1756)U100040 2562 256305 24946 (1353) 2562 256285 27351 (1238)1Best and average results from 100 runs2CPU seconds on a Pentium IV 28GHz3Average number of generations over 100 runs

heuristic but KL-MA andGEFM do notWe performed 1000runs for bipartitioning and 100 runs for 8-way and 32-waypartitioning All the programs were written in the C languageand compiled using GNUrsquos gcc compiler It was run on aPentium IV 28GHz computer with Linux In the tables thebold-faced numbers indicate the lower of the average cut sizes

obtained by the two algorithms CMPA outperformed KL-MA and GEFM on most graphs with comparable runningtimes

CMPA underperformed on some random graphs whichmay be due to the weak cluster structures in these graphsCMPArsquos average performance was better on all the geometric

8 Mathematical Problems in Engineering

Table 6 Comparison of GEFM and CMPA on 32-way partitioning

Graph Best known1 GEFM [21] CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

G50025 177 177 18169 7898 (1491) 177 18082 7430 (1382)G50005 623 624 63047 16226 (2197) 623 62992 15918 (2035)G50010 1573 1574 158194 30083 (2364) 1572lowast 157972 35573 (2332)G50020 4034 4037 404506 68153 (2565) 4035 404281 67319 (2479)G100025 312 313 32099 33579 (2798) 313 32025 35644 (2648)G100005 1217 1208 121872 69867 (3133) 1205lowast 121689 73806 (3203)G100010 3360 3353lowast 336731 117148 (3294) 3355 336608 132359 (3267)G100020 7818 7817 782981 201278 (3475) 7815lowast 783048 236369 (3342)U50005 109 112 11639 13829 (1640) 109 11318 12115 (1595)U50010 523 531 53775 22552 (1512) 526 53184 16626 (1222)U50020 1825 1831 184139 28507 (1344) 1823lowast 183188 29947 (1197)U50040 5328 5364 538001 56174 (1381) 5348 536983 52315 (1331)U100005 117 118 12602 45103 (3137) 115lowast 12349 46435 (2967)U100010 577 576 58326 59901 (2599) 571lowast 57816 74350 (2603)U100020 2367 2375 239633 79877 (1963) 2373 238893 93043 (1848)U100040 7329 7399 741749 142178 (1634) 7382 740718 149310 (1573)1Best known values [20ndash22 24 33]2Best and average results from 100 runs Asterisked numbers are new best values3CPU seconds on a Pentium IV 28GHz4Average number of generations from 100 runs

graphs This suggests that effective cluster handling is moreimportant on the geometric graphs as we suggested inSection 31

The local optimization algorithm ismuchmore expensivethan the cluster-handling heuristic thus CMPA does nottake much longer to run than KL-MA or GEFM On averageCMPA required 1414more time than KL-MA for the bipar-titioning problems and 202 more than GEFM in 32-waypartitioning CMPA ran 552 faster in 8-way partitioning

7 Concluding Remarks

We devised a cluster-moving heuristic and combined it witha memetic algorithm (MA) for graph partitioning Experi-ments on 2-way 8-way and 32-way partitioning showed thatthis heuristic significantly improved the performance of theMA especially on the 32-way partitioning

The method of moving clusters that we have introducedaddresses a significant weakness in standard IIP algorithmsThe idea of adding an operation to complement a localoptimization algorithm might be used to address otherdeficiencies in MAs

Our method of cluster detection is based on a measurethat we call genic distance which is designed to reflect theextent to which two vertices connected by an edge belongto the same cluster Instead of computing genic distancesonce in an initialization phase anMA could recompute thesedistances as it progresses this might improve the accuracyof cluster detection at some cost in time We believe thatgenic distance might also be useful in solving other problemsinvolving clusters such as the maximum clique problem

Table 7 Real-world benchmark graphs

Graph Number of vertices Number of edgesnasa4704 4704 50026bcspwr09 1723 2394bcsstk13 2003 40940DEBR12 4096 8189

Appendix

Results on Real-World Graphs

We also tested on four real-world benchmark graphs usedin [11 15 43] The sizes of the graphs are given in Table 7We conducted experiments on 2-way and 8-way partitioningWe performed 100 runs for bipartitioning and 50 runs for8-way partitioning It was run on an Intel Core i7 36GHzcomputer with Linux Table 8 compares CMPA with KL-MAon bipartitioning and Table 9 compares CMPA with GEFMon 8-way partitioning In the tables the bold-faced numbersindicate the lower of the average cut sizes obtained by the twoalgorithms Similarly to the results in Section 6CMPAoverallperformed better than the others with comparable runningtimes

Disclosure

Apreliminary version of this paper appeared inProceedings ofthe Genetic and Evolutionary Computation Conference 2007(p 1520)

Mathematical Problems in Engineering 9

Table 8 Comparison of KL-MA and CMPA on bipartitioning

Graph Best known1 KL-MA CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

nasa4704 1292 1292 129200 415 (407) 1292 129200 382 (347)bcspwr09 9 9 1131 061 (1010) 9 1060 081 (1003)bcsstk13 2355 2355 235500 117 (330) 2355 235500 146 (282)DEBR12 548 548 54844 991 (765) 548 54820 1032 (794)1Best result from the literature2Best and average results from 100 runs3CPU seconds on Intel Core i7 36GHz4Average number of generations over 100 runs

Table 9 Comparison of GEFM and CMPA on 8-way partitioning

Graph GEFM [21] CMPABest1 Ave1 CPU2 (Gen3) Best1 Ave1 CPU2 (Gen3)

nasa4704 3898 390366 57569 (2235) 3896 390266 59151 (2255)bcspwr09 53 5708 7880 (2947) 54 5774 8516 (2925)bcsstk13 8911 893910 28119 (1427) 8919 893614 30154 (1476)DEBR12 1248 126016 31597 (3969) 1248 125990 35383 (4002)1Best and average results from 50 runs2CPU seconds on Intel Core i7 36GHz3Average number of generations over 50 runs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the ICT RampD program ofMSIPIITP Korea (10045253 The development of SDNOpenFlow-Based Enterprise Network Controller Technol-ogy) project The authors would like to thank ProfessorByung-Ro Moon and Dr Yongjoo Song for their valuablesuggestions for improving this paper The authors also thankJong-Pil Kim for providing the source code of GEFM [21]

References

[1] Y Kanizo D Hay and I Keslassy ldquoPalette distributing tablesin software-defined networksrdquo in Proceedings of the IEEEInternational Conference on Computer Communications pp545ndash549 April 2013

[2] P Wette M Draxler and A Schwabe ldquoMaxiNet distributedemulation of software-defined networksrdquo in Proceedings of theIFIP Networking Conference pp 1ndash9 June 2014

[3] M R Garey and D S Johnson Computers and Intractability AGuide to the Theory of NP-Completeness W H Freeman 1979

[4] T N Bui and C Jones ldquoFinding good approximate vertex andedge partitions is NP-hardrdquo Information Processing Letters vol42 no 3 pp 153ndash159 1992

[5] Y-H Kim ldquoImproved implementation choices for iterativeimprovement partitioning algorithms on circuitsrdquo in Proceed-ings of the International Conference on Computer Design pp 30ndash34 July 2008

[6] Y Yoon and Y-H Kim ldquoNew bucket managements in iterativeimprovement partitioning algorithmsrdquo Applied Mathematics ampInformation Sciences vol 7 no 2 pp 529ndash532 2013

[7] BW Kernighan and S Lin ldquoAn efficient heuristic procedure forpartitioning graphsrdquo Bell System Technical Journal vol 49 no2 pp 291ndash307 1970

[8] C Fiduccia and R Mattheyses ldquoA Linear-time heuristic forimproving network partitionsrdquo in Proceedings of the Conferenceon Design Automation pp 175ndash181 June 1982

[9] D S Johnson C R Aragon L A McGeoch and C SchevonldquoOptimization by simulated annealing An experimental evalu-ation Part I Graph partitioningrdquo Operations Research vol 37no 6 pp 865ndash892 1989

[10] T N Bui and B R Moon ldquoGenetic algorithm and graphpartitioningrdquo IEEETransactions on Computers vol 45 no 7 pp841ndash855 1996

[11] R Battiti and A A Bertossi ldquoGreedy prohibition and reactiveheuristics for graph partitioningrdquo IEEE Transactions on Com-puters vol 48 no 4 pp 361ndash385 1999

[12] E Rolland H Pirkul and F Glover ldquoTabu search for graphpartitioningrdquo Annals of Operations Research vol 63 no 2 pp209ndash232 1996

[13] T N Bui and L C Strite ldquoAn ant system algorithm for graphbisectionrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference pp 43ndash51 July 2002

[14] I Hwang Y-H Kim and B-R Moon ldquoMulti-attractor genereordering for graph bisectionrdquo inProceedings of theGenetic andEvolutionary Computation Conference pp 1209ndash1215 July 2006

[15] Y-H Kim and B-R Moon ldquoLock gain based graph partition-ingrdquo Journal of Heuristics vol 10 no 1 pp 37ndash57 2004

[16] P Merz and B Freisleben ldquoFitness landscapes memetic algo-rithms and greedy operators for graph bipartitioningrdquo Evolu-tionary Computation vol 8 no 1 pp 61ndash91 2000

[17] HMuhlenbein and TMahnig ldquoEvolutionary optimization andthe estimation of search distributions with applications to graph

10 Mathematical Problems in Engineering

bipartitioningrdquo International Journal of Approximate Reasoningvol 31 no 3 pp 157ndash192 2002

[18] A G Steenbeek E Marchiori and A E Eiben ldquoFindingbalanced graph bi-partitions using a hybrid genetic algorithmrdquoin Proceedings of the IEEE International Conference on Evolu-tionary Computation pp 90ndash95 May 1998

[19] Y Yoon and Y-H Kim ldquoVertex ordering clustering andtheir application to graph partitioningrdquoApplied Mathematics ampInformation Sciences vol 8 no 1 pp 135ndash138 2014

[20] S-J Kang and B-R Moon ldquoA hybrid genetic algorithm formultiway graph partitioningrdquo in Proceedings of the Genetic andEvolutionary Computation Conference pp 159ndash166 July 2000

[21] J-P Kim and B-R Moon ldquoA hybrid genetic search for multi-way graph partitioning based on direct partitioningrdquo in Pro-ceedings of the Genetic and Evolutionary Computation Confer-ence pp 408ndash415 July 2001

[22] Y-H Kim A Moraglio Y Yoon and B-R Moon ldquoGeometriccrossover for multiway graph partitioningrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1217ndash1224 July 2006

[23] A Moraglio Y-H Kim Y Yoon and B-R Moon ldquoGeometriccrossovers for multiway graph partitioningrdquo Evolutionary Com-putation vol 15 no 4 pp 445ndash474 2007

[24] AMoraglio Y-H Kim Y Yoon B-RMoon and R Poli ldquoGeo-metric crossover for permutations with repetitions applicationto graph partitioningrdquo in Proceedings of the PPSN Workshopon Evolutionary AlgorithmsmdashBridging Theory and PracticeSeptember 2006

[25] A J Soper C Walshaw and M Cross ldquoA combined evolu-tionary search and multilevel optimisation approach to graph-partitioningrdquo Journal of Global Optimization vol 29 no 2 pp225ndash241 2004

[26] Y Yoon Y-H Kim A Moraglio and B-R Moon ldquoQuotientgeometric crossovers and redundant encodingsrdquo TheoreticalComputer Science vol 425 pp 4ndash16 2012

[27] J Kim I Hwang Y-H Kim and B-R Moon ldquoGeneticapproaches for graph partitioning a surveyrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 473ndash480 July 2011

[28] S Dutt and W Deng ldquoA probability-based approach to VLSIcircuit partitioningrdquo in Proceedings of the Design AutomationConference pp 100ndash105 June 1996

[29] B Krishnamurthy ldquoAn improved min-cut algorithm for parti-tioning VLSI networksrdquo IEEE Transactions on Computers vol33 no 5 pp 438ndash446 1984

[30] L A Sanchis ldquoMultiple-way network partitioningrdquo IEEETrans-actions on Computers vol 38 no 1 pp 62ndash81 1989

[31] G Laszewski ldquoIntelligent structural operators for the k-waygraph partitioning problemrdquo in Proceedings of the InternationalConference on Genetic Algorithms pp 45ndash52 July 1991

[32] H Muhlenbein ldquoParallel genetic algorithms in combinatorialoptimizationrdquo in Computer Science and Operations Research OBalci R Sharda and S Zenios Eds pp 441ndash453 PergamonOxford UK 1992

[33] S-S Choi and B-R Moon ldquoNormalization in genetic algo-rithmsrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference vol 2723 of Lecture Notes in ComputerScience pp 862ndash873 2003

[34] S-H Kim Y-H Kim and B-R Moon ldquoA hybrid geneticalgorithm for themax cut problemrdquo inProceedings of theGeneticand Evolutionary Computation Conference pp 416ndash423 July2001

[35] Y-H Kim Y-K Kwon and B-R Moon ldquoProblem-independ-ent schema synthesis for genetic algorithmsrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1112ndash1122 July 2003

[36] K Seo S Hyun and Y-H Kim ldquoAn edge-set representationbased on spanning tree for searching cut spacerdquo IEEE Trans-actions on Evolutionary Computation 2014

[37] Y-H Kim Combinatorial optimization based on effective localsearch genetic algorithms and problem space analyses [PhDthesis] Seoul National University Seoul Republic of Korea2005

[38] Y-H Kim ldquoAn enzyme-inspired approach to surmount barriersin graph bisectionrdquo in Proceedings of the International Confer-ence on Computational Science and Its Applications vol 5072 ofLecture Notes in Computer Science pp 841ndash851 2008

[39] S Dutt and W Deng ldquoVLSI circuit partitioning by cluster-removal using iterative improvement techniquesrdquo in Proceed-ings of the IEEEACM International Conference on Computer-Aided Design pp 194ndash200 November 1996

[40] S Dutt and W Deng ldquoCluster-aware iterative improvementtechniques for partitioning large VLSI circuitsrdquo ACM Transac-tions on Design Automation of Electronic Systems vol 7 no 1pp 91ndash121 2002

[41] Y Yoon and Y-H Kim ldquoAn efficient genetic algorithm formaximum coverage deployment in wireless sensor networksrdquoIEEE Transactions on Cybernetics vol 43 no 5 pp 1473ndash14832013

[42] Y Yoon Y-H Kim AMoraglio and B-RMoon ldquoA theoreticaland empirical study on unbiased boundary-extended crossoverfor real-valued representationrdquo Information Sciences vol 183no 1 pp 48ndash65 2012

[43] BMonien andRDiekmann ldquoA local graph partitioning heuris-tic meeting bisection boundsrdquo in Proceedings of the SIAM Con-ference on Parallel Processing for Scientific Computing March1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Cluster

3

4

1

2

5 edges connecting subsets

(a) Before moving a cluster

Cluster

3

4

1

2

1 edge connecting subsets

(b) After moving a cluster

Figure 1 An example of cluster moving in which the cut size of apartition is reduced by 4

graphs G119899119889 and 8 random geometric graphs U119899119889 The twodifferent classes of graphs are briefly described below

(i) G119899119889 a random graph on 119899 vertices with an indepen-dent probability 119901 that any two vertices are connectedby an edge The probability 119901 is biased so that theexpected vertex degree 119901(119899 minus 1) is 119889

(ii) U119899119889 a random geometric graph on 119899 vertices that liein the unit square and whose coordinates are chosenuniformly from the unit interval Every pair of ver-tices separated by a distance of 119905 or less is connectedby an edge The expected degree of a vertex is 1198991205871199052

3 Difficulty of Moving Clusters

Suppose that the cluster shown in Figure 1(a) is involved ina bipartitioning problem The four vertices in this cluster arefully interconnected and they all belong to the same subsetMoving this cluster to the other subset across the dottedline in Figure 1(b) will reduce the cut size of the partition by4 However there is no motivation to move any single vertexbecause they all have negative gain the gains of V

1 V2 V3 and

V4are minus1 minus4 minus2 and minus1 respectivelyThis example illustrates

how IIP algorithms may miss a significant reduction incut size that could be achieved by moving several verticestogether

The baleful effect of clusters on local search algorithmstrying to solve the graph partitioning problemmotivated thisstudy Kim [37 38] indicated that graph partitioning is hardprimarily due to the difficulty of moving clusters Dutt andDeng [39 40] have also observed that an IIP method appliedto circuit partitioning can fail because of the difficulty ofdealing with clusters that straddle subsets

31 Experimental Support We designed experiments toquantify the effect of clusters on IIP algorithms representedby the KL algorithm Using the cluster detection method tobe described in Section 41 we find clusters in the graph andselect one randomly We then take a locally optimum bipar-tition 119904 obtained by KL and move the selected cluster to theother subset creating a perturbed partition 119905cluster ApplyingKL to 119905cluster we obtain a new local optimum 119906cluster

Table 1 Probability that KL fails to return vertices moved from onesubset of a partition to the other when the vertices are in cluster(119875cluster) or chosen at random (119875random) over 1000 runs

Graph 119875cluster () 119875random () Average number of verticesmoved (119902)

G50025 1690 250 641

G50005 570 130 550

G50010 190 030 505

G50020 050 010 478

G100025 1280 190 589

G100005 290 080 561

G100010 060 020 467

G100020 030 000 491

U50005 3910 100 741

U50010 2620 050 1205

U50020 860 020 1410

U50040 1040 040 4936

U100005 3790 080 781

U100010 2850 060 1247

U100020 1160 010 2219

U100040 910 040 3588

Assuming that 119902 the number of vertices in the clusters issmall 119906cluster can be expected to match 119904 if KL is successfulin moving the cluster back However if KL fails to returnthe perturbed cluster to its original subset the cut size of thepartition may increase Repeating this experiment we derive119875cluster as an approximation of the probability that the cut sizeof 119906cluster is larger than that of 119904

For comparison we perturbed 119902 vertices randomlyselected within a locally optimized partition bymoving themto the other subset We call this partition 119905random We applythe KL algorithm to 119905random and then obtain a new locallyoptimum partition 119906random Repeating this experiment wederive 119875random as an approximation of the probability that thecut size of 119906random is larger than that of 119904

Table 1 shows the values of 119875cluster and 119875random for 16benchmark graphs We see that 119875cluster is always larger than119875random as we would expect We notice that the gap between119875cluster and 119875random is larger on the geometric graphs (U119899119889)than on the random graphs (G119899119889)

4 Cluster-Handling Heuristic

Cluster analysis is a well-known problem for which plenty ofalgorithms exist many of which require a lot of computationThe insight that motivates our heuristic is that the applicationof a local optimization process such as IIP to a randomly par-titioned graph creates a modified partition in which clusterstend to be wholly allocated to one subset or another (and arethen difficult to move as we have already observed) A singlepartition of this sort is of little help in identifying clustersbecause the clusters are not separated at all within eachsubset but if we create many random partitions and optimize

4 Mathematical Problems in Engineering

Table 2 Truth table for Fact 1

119901 119902 119901 rarr 119902 119868(119901) 119868(119902)

True True True 1 1

True False False 1 0

False True True 0 1

False False True 0 0

them we can reasonably infer that vertices that find them-selves in the same subset in most of these partitions belong tothe same cluster We can make this inference in a structuredway using the ldquogenic distancerdquo metric that we will introduceThis approach to cluster analysis may seem indirect but it isefficient in the context of an evolutionary approach to graphpartitioning because the set of partitions required for findingclusters using genic distance is also the population which wemust create to be evolved by our MA

One way of dealing with clusters is to devise a local opti-mization heuristic that can identify clusters [18 19 38] How-ever this prevents us from building on previous studies of IIPalgorithms

Our approach is to add an additional heuristic to ourMA which finds and moves clusters The heuristic identifiesclusters in the population of partitions which have alreadybeen optimized locally It selects clusters with higher gainsandmoves them IIP local optimization is then applied again

41 Cluster Detection Let 119868(sdot) be an indicator function that is119868(true) = 1 and 119868(false) = 0 Then we can trivially establishthe following

Fact 1 If 119901 rarr 119902 is true then 119868(119901) le 119868(119902)

Proof From Table 2

Fact 2 119868(119901 or 119902) le 119868(119901) + 119868(119902)

Proof From Table 3

We now define a metric called genic distance which mea-sure the extent to which two vertices connected by an edgecan be considered to belong to the same cluster We denotethe genic distance of an edge V

119894 V119895 within a population

1199011 1199012 119901

119898 of locally optimized partitions as 119889

119892(V119894 V119895)

which can be expressed as follows

119889119892(V119894 V119895) =

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119895)) (3)

where 119892119894and 119892

119895are the genes corresponding to V

119894and V

119895

respectively The value of gene 119892119894(ie the partition number

that vertex V119894belongs to) in the 119897th individual is 119901

119897(119892119894) For

convenience we assume that each vertex has an edge thatconnects it to itself so that V V isin 119864 for each vertex V Thenthe following proposition holds

Proposition 1 For each population 119889119892becomes a pseudomet-

ric on 119881

Table 3 Truth table for Fact 2

119901 119902 119868(119901) 119868(119902) 119901 or 119902 119868(119901 or 119902) 119868(119901) + 119868(119902)True True 1 1 True 1 2

True False 1 0 True 1 1

False True 0 1 True 1 1

False False 0 0 False 0 0

Proof Since 119868(sdot) ge 0 119889119892(V119894 V119895) = sum

119898

119897=1119868(119901119897(119892119894) = 119901119897(119892119895)) ge 0

for each V119894 V119895 isin 119864 It is enough to show the following three

conditions

(i) 119889119892(V119894 V119894) = 0

119889119892(V119894 V119894) =

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119894))

=

119898

sum

119897=1

119868 (false)

= 0

(4)

(ii) Symmetry Let V119894 V119895 be in 119864

119889119892(V119894 V119895) =

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119895))

=

119898

sum

119897=1

119868 (119901119897(119892119895) = 119901119897(119892119894))

= 119889119892(V119895 V119894)

(5)

(iii) Triangle Inequality Consider each group of threeedges V

119894 V119895 V119894 V119896 V119896 V119895 isin 119864 If 119901

119897(119892119894) = 119901

119897(119892119896)

and119901119897(119892119896) = 119901119897(119892119895) then119901

119897(119892119894) = 119901119897(119892119895) for each 119897 By

contraposition if 119901119897(119892119894) = 119901119897(119892119895) then 119901

119897(119892119894) = 119901119897(119892119896)

or 119901119897(119892119896) = 119901119897(119892119895)

For each 119897

119868 (119901119897(119892119894) = 119901119897(119892119895))

le 119868 (119901119897(119892119894) = 119901119897(119892119896) or 119901119897(119892119896) = 119901119897(119892119895)) (∵ Fact 1)

le 119868 (119901119897(119892119894) = 119901119897(119892119896)) + 119868 (119901

119897(119892119896) = 119901119897(119892119895)) (∵ Fact 2)

(6)

By summing the above inequalities for all 119897

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119895))

le

119898

sum

119897=1

(119868 (119901119897(119892119894) = 119901119897(119892119896)) + 119868 (119901

119897(119892119896) = 119901119897(119892119895)))

=

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119896)) +

119898

sum

119897=1

119868 (119901119897(119892119896) = 119901119897(119892119895))

(7)

Mathematical Problems in Engineering 5

Therefore we have

119889119892(V119894 V119895) le 119889119892(V119894 V119896) + 119889119892(V119896 V119895) (8)

That is 119889119892satisfies the triangle inequality

Proposition 1 suggests that the measure 119889119892is reasonable

A pseudometric space is a generalization of a metric space inwhich points need not be distinguishable thus it is possiblethat 119889

119892(V119894 V119895) = 0 for some edge V

119894 V119895 with distinct vertices

V119894= V119895

Our heuristic detects clusters by collecting a number oflocal optima and computes genic distances for all the edges ineach graphThis takes119874(119898|119864|) time but this cost is negligiblesince this computation is a preprocess performed before theMA runs The heuristic temporarily eliminates edges withgenic distances that are greater than a threshold value 120579 Weset 120579 to be the smallest value that satisfies

10038161003816100381610038161003816119890 isin 119864 119889

119892(119890) le 120579

10038161003816100381610038161003816le 01 |119881| (9)

Each remaining connected component containingmore thanthree vertices is considered to be a cluster

Figure 2 shows how our heuristic detects clusters Fig-ure 2(a) shows an example graph with 11 vertices and 15edges Four individuals corresponding to locally optimizedpartitions from the population are shown in Figure 2(b)In Figure 2(c) each edge is labeled with its genic distanceIf the threshold value of genic distance is 1 then theedges with larger genic distances indicated by dotted lineare eliminated Then four connected components remainV3 V7 V8 V9 V2 V4 V6 V11 V1 V5 and V

10 The last two

of these connected components are considered too small tobe clusters Thus clusters V

3 V7 V8 V9 and V

2 V4 V6 V11

shaded in Figure 2(c) remain as candidates for moving

42 Cluster-Moving Scheme Our heuristic improves theoffspring of each generation after crossover by moving theclusters that were detected using the technique described inthe previous subsection To select the clusters to be movedand their target subsets we introduce ameasure called clustergain such that cg(119909 119886) is the reduction in the cut size of thepartition when all the vertices in cluster 119909 are moved to thesubset 119886 For example moving the cluster in Figure 1(a) to theother subset in the partition is associated with a cluster gainof 2

This cluster-moving scheme described in Algorithm 1is applied to each individual generated by crossover whichis a partition that may be unbalanced However cut sizeand cluster gain are well defined on unbalanced partitionsOur scheme does not consider moving every cluster in everypartition because we found that making all clusters movablecauses the premature convergence of the MA Thus ourheuristic selects119873 clusters at random as candidates for mov-ing In our experiments we set119873 to 5We compute the clustergain that results from moving each candidate cluster to eachof the other 119896 minus 1 subsets The candidate cluster 119909 anddestination subset 119886with the highest cluster gain are selectedAssume that cg(119909 119886) is positive all the vertices in cluster 119909

6

12

3 45

78

9 1011

(a) Example graph

1

1 1 1 1 1 1

1 113 5 7 8 11

1

1111110 0 0 0 0

10 0

0 0 0 0 0

1 1 1 1 10 00 0 0 0

0 0 0 010642 9

(b) Four locally optimized partitionsin the population (subsets 0 and 1)

00

00

0

00

11

1

23

03

3

(c) Two detected clusters

Figure 2 An example of cluster detection

are moved to subset 119886 and cluster 119909 is removed from theset of candidate clusters This process is repeated until nocandidates remain or no move yields a positive cluster gainNo attempt is made to balance the partition during cluster-moving this is performed later

5 Memetic Search Framework

CMPA (cluster-moving memetic partitioning algorithm) is amemetic algorithm that we have designed for graph parti-tioning In this MA an individual is a 119896-way partition Eachgene in an individual corresponds to a vertex and has a valuebetween 0 and 119896 minus 1 which indicates the subset to which thevertex belongs that is the 119894th gene 119892

119894= 119895 hArr V

119894isin 119875119895+1

It isa steady-state MA meaning that there is only one offspringfrom population in each generation Crossover is followed bya cluster-moving step and then local optimization

Algorithm 2 shows the processes that make up CMPAwhich we now describe in detail

(i) Initialization When the MA starts 119898 individuals(ie partitions) are created at random Then eachindividual is improved by local optimization We set119898 to be 30 for bipartitioning and 50 for multiwaypartitioning (with 119896 = 8 or 119896 = 32)

6 Mathematical Problems in Engineering

Select clusters 119878 larr 119873 clusters selected at randomdo

Calculate cg(119909 119886) for all 119909 isin 119878 1 le 119886 le 119896cglowast larr max

119909119886cg(119909 119886)

(119909 119886) larr argmax119909119886

cg(119909 119886)

if cglowast gt 0 then

Move cluster 119909 to partition 119886119878 larr 119878 119909

until (119878 = 0 or cglowast le 0)

Algorithm 1 Our cluster-moving scheme

Create initial population of fixed sizeApply local optimization to each member of populationCalculate genic distance from populationFind clusters and store their informationdo

Select 1199011198861199031198901198991199051and 119901119886119903119890119899119905

2from population

Normalization(1199011198861199031198901198991199051 119901119886119903119890119899119905

2)

offspring larr Crossover(1199011198861199031198901198991199051 119901119886119903119890119899119905

2)

Cluster-moving(offspring)Local-optimization(offspring)Replace(population offspring)

until (stopping condition)return the best solution

Algorithm 2 The process in CMPA

(ii) Selection We used the roulette-wheel-based propor-tional selection The probability that the best individ-ual is chosen was set to four times the probabilitythat the worst is chosen The fitness value 119891

119894of the

119894th individual is expressed as (119888119908minus 119888119894) + (119888119908minus 119888119887)3

where 119888119887 119888119908 and 119888

119894are the cut sizes of the partitions

corresponding to the best the worst and the 119894thindividual respectively

(iii) Normalization Laszewski [31] first used normaliza-tion to improve the performance of GA and its vari-ants have been suggested in [23 26 41 42]The parentindividuals are normalized before crossover followingLaszewski [31 33] The subset of one parent whichshares the largest number of vertices with subsetsof the other parent is selected and that subset isnumbered 0 This process is repeated incrementingthe index until all subsets are numbered

(iv) Crossover and Cluster Moving We used a standardfive-point crossover After crossover the cluster-han-dling heuristic described in Section 42 is applied tothe individual At this point individuals usually cor-respond to unbalanced partitionsWe select a randomlocation in the individual and adjust the values ofthe genes which are the subsets to which the corre-sponding vertices belong to the right of this location(in a typical circular string) until the partition is

balanced This is effectively a mutation effect and nofurther mutation was introduced

(v) Local Optimization KL [7] was used for the bisectionproblems and EFM (extended FM) [21] was used formultiway partitioning (119896 = 8 or 119896 = 32)

(vi) Replacement We used a replacement scheme due toBui and Moon [10] If an offspring is better than itscloser parent the MA replaces that parent If it is bet-ter than its other parent then that parent is replacedOtherwise it replaces the worst individual in thepopulation

(vii) Stopping Condition This is based on consecutive fail-ures to replace an individualrsquos parents Termination istriggered by consecutive failures 30 in bipartitioningand 50 in multiway partitioning

6 Experimental Results

We conducted experiments on 2-way 8-way and 32-way par-titioning Table 4 shows the performance of MA combinedwith KL (denoted KL-MA) and CMPA on bipartitioningTable 5 shows the performance of the genetic extended FMalgorithm (GEFM) [21] one of themost effective approachesand CMPA on 8-way partitioning and Table 6 gives theresults for 32-way partitioning CMPAuses a cluster-handling

Mathematical Problems in Engineering 7

Table 4 Comparison of KL-MA and CMPA on bipartitioning

Graph Best known1 KL-MA CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

G50025 49 49 5134 032 (595) 49 5122 034 (558)G50005 218 218 21865 059 (700) 218 21845 064 (682)G50010 626 626 62755 102 (681) 626 62746 110 (672)G50020 1744 1744 174583 193 (694) 1744 174562 209 (687)G100025 93 93 9726 096 (745) 93 9708 104 (722)G100005 445 445 45150 207 (884) 445 45092 222 (860)G100010 1362 1362 136668 407 (969) 1362 136618 445 (977)G100020 3382 3382 338561 742 (971) 3382 338526 813 (987)U50005 2 2 496 055 (793) 2 394 066 (803)U50010 26 26 2631 045 (459) 26 2604 057 (465)U50020 178 178 17806 042 (300) 178 17800 053 (271)U50040 412 412 41200 034 (190) 412 41200 037 (102)U100005 1 1 1167 171 (1077) 1 881 212 (1068)U100010 39 39 4627 151 (645) 39 4415 195 (653)U100020 222 222 22219 113 (347) 222 22207 151 (327)U100040 737 737 73700 120 (243) 737 73700 177 (205)1Best result from the literature2Best and average results from 1000 runs3CPU seconds on a Pentium IV 28GHz4Average number of generations over 1000 runs

Table 5 Comparison of GEFM and CMPA on 8-way partitioning

Graph GEFM [21] CMPABest1 Ave1 CPU2 (Gen3) Best1 Ave1 CPU2 (Gen3)

G50025 111 11521 4203 (2090) 111 11451 4025 (2091)G50005 465 46816 8565 (2314) 465 46763 7296 (2239)G50010 1254 125953 14454 (2347) 1254 125847 13504 (2297)G50020 3348 335480 29318 (2437) 3348 335371 28509 (2317)G100025 212 21630 14676 (3043) 211 21657 14842 (3023)G100005 930 93811 24720 (3148) 931 93926 23467 (3069)G100010 2714 272633 40896 (3146) 2711 272552 41698 (3083)G100020 6525 653655 79129 (3303) 6520 653835 78167 (3141)U50005 16 1880 7087 (2629) 16 1722 5589 (2137)U50010 143 14512 9875 (2277) 143 14426 9050 (1984)U50020 612 61457 12122 (1690) 611 61293 9753 (1483)U50040 1867 187260 15794 (1396) 1867 187173 12962 (1204)U100005 21 3366 18035 (3638) 20 2853 20953 (3172)U100010 176 18370 24308 (3105) 176 18230 25702 (2857)U100020 812 81405 23108 (1824) 812 81300 23293 (1756)U100040 2562 256305 24946 (1353) 2562 256285 27351 (1238)1Best and average results from 100 runs2CPU seconds on a Pentium IV 28GHz3Average number of generations over 100 runs

heuristic but KL-MA andGEFM do notWe performed 1000runs for bipartitioning and 100 runs for 8-way and 32-waypartitioning All the programs were written in the C languageand compiled using GNUrsquos gcc compiler It was run on aPentium IV 28GHz computer with Linux In the tables thebold-faced numbers indicate the lower of the average cut sizes

obtained by the two algorithms CMPA outperformed KL-MA and GEFM on most graphs with comparable runningtimes

CMPA underperformed on some random graphs whichmay be due to the weak cluster structures in these graphsCMPArsquos average performance was better on all the geometric

8 Mathematical Problems in Engineering

Table 6 Comparison of GEFM and CMPA on 32-way partitioning

Graph Best known1 GEFM [21] CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

G50025 177 177 18169 7898 (1491) 177 18082 7430 (1382)G50005 623 624 63047 16226 (2197) 623 62992 15918 (2035)G50010 1573 1574 158194 30083 (2364) 1572lowast 157972 35573 (2332)G50020 4034 4037 404506 68153 (2565) 4035 404281 67319 (2479)G100025 312 313 32099 33579 (2798) 313 32025 35644 (2648)G100005 1217 1208 121872 69867 (3133) 1205lowast 121689 73806 (3203)G100010 3360 3353lowast 336731 117148 (3294) 3355 336608 132359 (3267)G100020 7818 7817 782981 201278 (3475) 7815lowast 783048 236369 (3342)U50005 109 112 11639 13829 (1640) 109 11318 12115 (1595)U50010 523 531 53775 22552 (1512) 526 53184 16626 (1222)U50020 1825 1831 184139 28507 (1344) 1823lowast 183188 29947 (1197)U50040 5328 5364 538001 56174 (1381) 5348 536983 52315 (1331)U100005 117 118 12602 45103 (3137) 115lowast 12349 46435 (2967)U100010 577 576 58326 59901 (2599) 571lowast 57816 74350 (2603)U100020 2367 2375 239633 79877 (1963) 2373 238893 93043 (1848)U100040 7329 7399 741749 142178 (1634) 7382 740718 149310 (1573)1Best known values [20ndash22 24 33]2Best and average results from 100 runs Asterisked numbers are new best values3CPU seconds on a Pentium IV 28GHz4Average number of generations from 100 runs

graphs This suggests that effective cluster handling is moreimportant on the geometric graphs as we suggested inSection 31

The local optimization algorithm ismuchmore expensivethan the cluster-handling heuristic thus CMPA does nottake much longer to run than KL-MA or GEFM On averageCMPA required 1414more time than KL-MA for the bipar-titioning problems and 202 more than GEFM in 32-waypartitioning CMPA ran 552 faster in 8-way partitioning

7 Concluding Remarks

We devised a cluster-moving heuristic and combined it witha memetic algorithm (MA) for graph partitioning Experi-ments on 2-way 8-way and 32-way partitioning showed thatthis heuristic significantly improved the performance of theMA especially on the 32-way partitioning

The method of moving clusters that we have introducedaddresses a significant weakness in standard IIP algorithmsThe idea of adding an operation to complement a localoptimization algorithm might be used to address otherdeficiencies in MAs

Our method of cluster detection is based on a measurethat we call genic distance which is designed to reflect theextent to which two vertices connected by an edge belongto the same cluster Instead of computing genic distancesonce in an initialization phase anMA could recompute thesedistances as it progresses this might improve the accuracyof cluster detection at some cost in time We believe thatgenic distance might also be useful in solving other problemsinvolving clusters such as the maximum clique problem

Table 7 Real-world benchmark graphs

Graph Number of vertices Number of edgesnasa4704 4704 50026bcspwr09 1723 2394bcsstk13 2003 40940DEBR12 4096 8189

Appendix

Results on Real-World Graphs

We also tested on four real-world benchmark graphs usedin [11 15 43] The sizes of the graphs are given in Table 7We conducted experiments on 2-way and 8-way partitioningWe performed 100 runs for bipartitioning and 50 runs for8-way partitioning It was run on an Intel Core i7 36GHzcomputer with Linux Table 8 compares CMPA with KL-MAon bipartitioning and Table 9 compares CMPA with GEFMon 8-way partitioning In the tables the bold-faced numbersindicate the lower of the average cut sizes obtained by the twoalgorithms Similarly to the results in Section 6CMPAoverallperformed better than the others with comparable runningtimes

Disclosure

Apreliminary version of this paper appeared inProceedings ofthe Genetic and Evolutionary Computation Conference 2007(p 1520)

Mathematical Problems in Engineering 9

Table 8 Comparison of KL-MA and CMPA on bipartitioning

Graph Best known1 KL-MA CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

nasa4704 1292 1292 129200 415 (407) 1292 129200 382 (347)bcspwr09 9 9 1131 061 (1010) 9 1060 081 (1003)bcsstk13 2355 2355 235500 117 (330) 2355 235500 146 (282)DEBR12 548 548 54844 991 (765) 548 54820 1032 (794)1Best result from the literature2Best and average results from 100 runs3CPU seconds on Intel Core i7 36GHz4Average number of generations over 100 runs

Table 9 Comparison of GEFM and CMPA on 8-way partitioning

Graph GEFM [21] CMPABest1 Ave1 CPU2 (Gen3) Best1 Ave1 CPU2 (Gen3)

nasa4704 3898 390366 57569 (2235) 3896 390266 59151 (2255)bcspwr09 53 5708 7880 (2947) 54 5774 8516 (2925)bcsstk13 8911 893910 28119 (1427) 8919 893614 30154 (1476)DEBR12 1248 126016 31597 (3969) 1248 125990 35383 (4002)1Best and average results from 50 runs2CPU seconds on Intel Core i7 36GHz3Average number of generations over 50 runs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the ICT RampD program ofMSIPIITP Korea (10045253 The development of SDNOpenFlow-Based Enterprise Network Controller Technol-ogy) project The authors would like to thank ProfessorByung-Ro Moon and Dr Yongjoo Song for their valuablesuggestions for improving this paper The authors also thankJong-Pil Kim for providing the source code of GEFM [21]

References

[1] Y Kanizo D Hay and I Keslassy ldquoPalette distributing tablesin software-defined networksrdquo in Proceedings of the IEEEInternational Conference on Computer Communications pp545ndash549 April 2013

[2] P Wette M Draxler and A Schwabe ldquoMaxiNet distributedemulation of software-defined networksrdquo in Proceedings of theIFIP Networking Conference pp 1ndash9 June 2014

[3] M R Garey and D S Johnson Computers and Intractability AGuide to the Theory of NP-Completeness W H Freeman 1979

[4] T N Bui and C Jones ldquoFinding good approximate vertex andedge partitions is NP-hardrdquo Information Processing Letters vol42 no 3 pp 153ndash159 1992

[5] Y-H Kim ldquoImproved implementation choices for iterativeimprovement partitioning algorithms on circuitsrdquo in Proceed-ings of the International Conference on Computer Design pp 30ndash34 July 2008

[6] Y Yoon and Y-H Kim ldquoNew bucket managements in iterativeimprovement partitioning algorithmsrdquo Applied Mathematics ampInformation Sciences vol 7 no 2 pp 529ndash532 2013

[7] BW Kernighan and S Lin ldquoAn efficient heuristic procedure forpartitioning graphsrdquo Bell System Technical Journal vol 49 no2 pp 291ndash307 1970

[8] C Fiduccia and R Mattheyses ldquoA Linear-time heuristic forimproving network partitionsrdquo in Proceedings of the Conferenceon Design Automation pp 175ndash181 June 1982

[9] D S Johnson C R Aragon L A McGeoch and C SchevonldquoOptimization by simulated annealing An experimental evalu-ation Part I Graph partitioningrdquo Operations Research vol 37no 6 pp 865ndash892 1989

[10] T N Bui and B R Moon ldquoGenetic algorithm and graphpartitioningrdquo IEEETransactions on Computers vol 45 no 7 pp841ndash855 1996

[11] R Battiti and A A Bertossi ldquoGreedy prohibition and reactiveheuristics for graph partitioningrdquo IEEE Transactions on Com-puters vol 48 no 4 pp 361ndash385 1999

[12] E Rolland H Pirkul and F Glover ldquoTabu search for graphpartitioningrdquo Annals of Operations Research vol 63 no 2 pp209ndash232 1996

[13] T N Bui and L C Strite ldquoAn ant system algorithm for graphbisectionrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference pp 43ndash51 July 2002

[14] I Hwang Y-H Kim and B-R Moon ldquoMulti-attractor genereordering for graph bisectionrdquo inProceedings of theGenetic andEvolutionary Computation Conference pp 1209ndash1215 July 2006

[15] Y-H Kim and B-R Moon ldquoLock gain based graph partition-ingrdquo Journal of Heuristics vol 10 no 1 pp 37ndash57 2004

[16] P Merz and B Freisleben ldquoFitness landscapes memetic algo-rithms and greedy operators for graph bipartitioningrdquo Evolu-tionary Computation vol 8 no 1 pp 61ndash91 2000

[17] HMuhlenbein and TMahnig ldquoEvolutionary optimization andthe estimation of search distributions with applications to graph

10 Mathematical Problems in Engineering

bipartitioningrdquo International Journal of Approximate Reasoningvol 31 no 3 pp 157ndash192 2002

[18] A G Steenbeek E Marchiori and A E Eiben ldquoFindingbalanced graph bi-partitions using a hybrid genetic algorithmrdquoin Proceedings of the IEEE International Conference on Evolu-tionary Computation pp 90ndash95 May 1998

[19] Y Yoon and Y-H Kim ldquoVertex ordering clustering andtheir application to graph partitioningrdquoApplied Mathematics ampInformation Sciences vol 8 no 1 pp 135ndash138 2014

[20] S-J Kang and B-R Moon ldquoA hybrid genetic algorithm formultiway graph partitioningrdquo in Proceedings of the Genetic andEvolutionary Computation Conference pp 159ndash166 July 2000

[21] J-P Kim and B-R Moon ldquoA hybrid genetic search for multi-way graph partitioning based on direct partitioningrdquo in Pro-ceedings of the Genetic and Evolutionary Computation Confer-ence pp 408ndash415 July 2001

[22] Y-H Kim A Moraglio Y Yoon and B-R Moon ldquoGeometriccrossover for multiway graph partitioningrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1217ndash1224 July 2006

[23] A Moraglio Y-H Kim Y Yoon and B-R Moon ldquoGeometriccrossovers for multiway graph partitioningrdquo Evolutionary Com-putation vol 15 no 4 pp 445ndash474 2007

[24] AMoraglio Y-H Kim Y Yoon B-RMoon and R Poli ldquoGeo-metric crossover for permutations with repetitions applicationto graph partitioningrdquo in Proceedings of the PPSN Workshopon Evolutionary AlgorithmsmdashBridging Theory and PracticeSeptember 2006

[25] A J Soper C Walshaw and M Cross ldquoA combined evolu-tionary search and multilevel optimisation approach to graph-partitioningrdquo Journal of Global Optimization vol 29 no 2 pp225ndash241 2004

[26] Y Yoon Y-H Kim A Moraglio and B-R Moon ldquoQuotientgeometric crossovers and redundant encodingsrdquo TheoreticalComputer Science vol 425 pp 4ndash16 2012

[27] J Kim I Hwang Y-H Kim and B-R Moon ldquoGeneticapproaches for graph partitioning a surveyrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 473ndash480 July 2011

[28] S Dutt and W Deng ldquoA probability-based approach to VLSIcircuit partitioningrdquo in Proceedings of the Design AutomationConference pp 100ndash105 June 1996

[29] B Krishnamurthy ldquoAn improved min-cut algorithm for parti-tioning VLSI networksrdquo IEEE Transactions on Computers vol33 no 5 pp 438ndash446 1984

[30] L A Sanchis ldquoMultiple-way network partitioningrdquo IEEETrans-actions on Computers vol 38 no 1 pp 62ndash81 1989

[31] G Laszewski ldquoIntelligent structural operators for the k-waygraph partitioning problemrdquo in Proceedings of the InternationalConference on Genetic Algorithms pp 45ndash52 July 1991

[32] H Muhlenbein ldquoParallel genetic algorithms in combinatorialoptimizationrdquo in Computer Science and Operations Research OBalci R Sharda and S Zenios Eds pp 441ndash453 PergamonOxford UK 1992

[33] S-S Choi and B-R Moon ldquoNormalization in genetic algo-rithmsrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference vol 2723 of Lecture Notes in ComputerScience pp 862ndash873 2003

[34] S-H Kim Y-H Kim and B-R Moon ldquoA hybrid geneticalgorithm for themax cut problemrdquo inProceedings of theGeneticand Evolutionary Computation Conference pp 416ndash423 July2001

[35] Y-H Kim Y-K Kwon and B-R Moon ldquoProblem-independ-ent schema synthesis for genetic algorithmsrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1112ndash1122 July 2003

[36] K Seo S Hyun and Y-H Kim ldquoAn edge-set representationbased on spanning tree for searching cut spacerdquo IEEE Trans-actions on Evolutionary Computation 2014

[37] Y-H Kim Combinatorial optimization based on effective localsearch genetic algorithms and problem space analyses [PhDthesis] Seoul National University Seoul Republic of Korea2005

[38] Y-H Kim ldquoAn enzyme-inspired approach to surmount barriersin graph bisectionrdquo in Proceedings of the International Confer-ence on Computational Science and Its Applications vol 5072 ofLecture Notes in Computer Science pp 841ndash851 2008

[39] S Dutt and W Deng ldquoVLSI circuit partitioning by cluster-removal using iterative improvement techniquesrdquo in Proceed-ings of the IEEEACM International Conference on Computer-Aided Design pp 194ndash200 November 1996

[40] S Dutt and W Deng ldquoCluster-aware iterative improvementtechniques for partitioning large VLSI circuitsrdquo ACM Transac-tions on Design Automation of Electronic Systems vol 7 no 1pp 91ndash121 2002

[41] Y Yoon and Y-H Kim ldquoAn efficient genetic algorithm formaximum coverage deployment in wireless sensor networksrdquoIEEE Transactions on Cybernetics vol 43 no 5 pp 1473ndash14832013

[42] Y Yoon Y-H Kim AMoraglio and B-RMoon ldquoA theoreticaland empirical study on unbiased boundary-extended crossoverfor real-valued representationrdquo Information Sciences vol 183no 1 pp 48ndash65 2012

[43] BMonien andRDiekmann ldquoA local graph partitioning heuris-tic meeting bisection boundsrdquo in Proceedings of the SIAM Con-ference on Parallel Processing for Scientific Computing March1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

Table 2 Truth table for Fact 1

119901 119902 119901 rarr 119902 119868(119901) 119868(119902)

True True True 1 1

True False False 1 0

False True True 0 1

False False True 0 0

them we can reasonably infer that vertices that find them-selves in the same subset in most of these partitions belong tothe same cluster We can make this inference in a structuredway using the ldquogenic distancerdquo metric that we will introduceThis approach to cluster analysis may seem indirect but it isefficient in the context of an evolutionary approach to graphpartitioning because the set of partitions required for findingclusters using genic distance is also the population which wemust create to be evolved by our MA

One way of dealing with clusters is to devise a local opti-mization heuristic that can identify clusters [18 19 38] How-ever this prevents us from building on previous studies of IIPalgorithms

Our approach is to add an additional heuristic to ourMA which finds and moves clusters The heuristic identifiesclusters in the population of partitions which have alreadybeen optimized locally It selects clusters with higher gainsandmoves them IIP local optimization is then applied again

41 Cluster Detection Let 119868(sdot) be an indicator function that is119868(true) = 1 and 119868(false) = 0 Then we can trivially establishthe following

Fact 1 If 119901 rarr 119902 is true then 119868(119901) le 119868(119902)

Proof From Table 2

Fact 2 119868(119901 or 119902) le 119868(119901) + 119868(119902)

Proof From Table 3

We now define a metric called genic distance which mea-sure the extent to which two vertices connected by an edgecan be considered to belong to the same cluster We denotethe genic distance of an edge V

119894 V119895 within a population

1199011 1199012 119901

119898 of locally optimized partitions as 119889

119892(V119894 V119895)

which can be expressed as follows

119889119892(V119894 V119895) =

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119895)) (3)

where 119892119894and 119892

119895are the genes corresponding to V

119894and V

119895

respectively The value of gene 119892119894(ie the partition number

that vertex V119894belongs to) in the 119897th individual is 119901

119897(119892119894) For

convenience we assume that each vertex has an edge thatconnects it to itself so that V V isin 119864 for each vertex V Thenthe following proposition holds

Proposition 1 For each population 119889119892becomes a pseudomet-

ric on 119881

Table 3 Truth table for Fact 2

119901 119902 119868(119901) 119868(119902) 119901 or 119902 119868(119901 or 119902) 119868(119901) + 119868(119902)True True 1 1 True 1 2

True False 1 0 True 1 1

False True 0 1 True 1 1

False False 0 0 False 0 0

Proof Since 119868(sdot) ge 0 119889119892(V119894 V119895) = sum

119898

119897=1119868(119901119897(119892119894) = 119901119897(119892119895)) ge 0

for each V119894 V119895 isin 119864 It is enough to show the following three

conditions

(i) 119889119892(V119894 V119894) = 0

119889119892(V119894 V119894) =

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119894))

=

119898

sum

119897=1

119868 (false)

= 0

(4)

(ii) Symmetry Let V119894 V119895 be in 119864

119889119892(V119894 V119895) =

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119895))

=

119898

sum

119897=1

119868 (119901119897(119892119895) = 119901119897(119892119894))

= 119889119892(V119895 V119894)

(5)

(iii) Triangle Inequality Consider each group of threeedges V

119894 V119895 V119894 V119896 V119896 V119895 isin 119864 If 119901

119897(119892119894) = 119901

119897(119892119896)

and119901119897(119892119896) = 119901119897(119892119895) then119901

119897(119892119894) = 119901119897(119892119895) for each 119897 By

contraposition if 119901119897(119892119894) = 119901119897(119892119895) then 119901

119897(119892119894) = 119901119897(119892119896)

or 119901119897(119892119896) = 119901119897(119892119895)

For each 119897

119868 (119901119897(119892119894) = 119901119897(119892119895))

le 119868 (119901119897(119892119894) = 119901119897(119892119896) or 119901119897(119892119896) = 119901119897(119892119895)) (∵ Fact 1)

le 119868 (119901119897(119892119894) = 119901119897(119892119896)) + 119868 (119901

119897(119892119896) = 119901119897(119892119895)) (∵ Fact 2)

(6)

By summing the above inequalities for all 119897

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119895))

le

119898

sum

119897=1

(119868 (119901119897(119892119894) = 119901119897(119892119896)) + 119868 (119901

119897(119892119896) = 119901119897(119892119895)))

=

119898

sum

119897=1

119868 (119901119897(119892119894) = 119901119897(119892119896)) +

119898

sum

119897=1

119868 (119901119897(119892119896) = 119901119897(119892119895))

(7)

Mathematical Problems in Engineering 5

Therefore we have

119889119892(V119894 V119895) le 119889119892(V119894 V119896) + 119889119892(V119896 V119895) (8)

That is 119889119892satisfies the triangle inequality

Proposition 1 suggests that the measure 119889119892is reasonable

A pseudometric space is a generalization of a metric space inwhich points need not be distinguishable thus it is possiblethat 119889

119892(V119894 V119895) = 0 for some edge V

119894 V119895 with distinct vertices

V119894= V119895

Our heuristic detects clusters by collecting a number oflocal optima and computes genic distances for all the edges ineach graphThis takes119874(119898|119864|) time but this cost is negligiblesince this computation is a preprocess performed before theMA runs The heuristic temporarily eliminates edges withgenic distances that are greater than a threshold value 120579 Weset 120579 to be the smallest value that satisfies

10038161003816100381610038161003816119890 isin 119864 119889

119892(119890) le 120579

10038161003816100381610038161003816le 01 |119881| (9)

Each remaining connected component containingmore thanthree vertices is considered to be a cluster

Figure 2 shows how our heuristic detects clusters Fig-ure 2(a) shows an example graph with 11 vertices and 15edges Four individuals corresponding to locally optimizedpartitions from the population are shown in Figure 2(b)In Figure 2(c) each edge is labeled with its genic distanceIf the threshold value of genic distance is 1 then theedges with larger genic distances indicated by dotted lineare eliminated Then four connected components remainV3 V7 V8 V9 V2 V4 V6 V11 V1 V5 and V

10 The last two

of these connected components are considered too small tobe clusters Thus clusters V

3 V7 V8 V9 and V

2 V4 V6 V11

shaded in Figure 2(c) remain as candidates for moving

42 Cluster-Moving Scheme Our heuristic improves theoffspring of each generation after crossover by moving theclusters that were detected using the technique described inthe previous subsection To select the clusters to be movedand their target subsets we introduce ameasure called clustergain such that cg(119909 119886) is the reduction in the cut size of thepartition when all the vertices in cluster 119909 are moved to thesubset 119886 For example moving the cluster in Figure 1(a) to theother subset in the partition is associated with a cluster gainof 2

This cluster-moving scheme described in Algorithm 1is applied to each individual generated by crossover whichis a partition that may be unbalanced However cut sizeand cluster gain are well defined on unbalanced partitionsOur scheme does not consider moving every cluster in everypartition because we found that making all clusters movablecauses the premature convergence of the MA Thus ourheuristic selects119873 clusters at random as candidates for mov-ing In our experiments we set119873 to 5We compute the clustergain that results from moving each candidate cluster to eachof the other 119896 minus 1 subsets The candidate cluster 119909 anddestination subset 119886with the highest cluster gain are selectedAssume that cg(119909 119886) is positive all the vertices in cluster 119909

6

12

3 45

78

9 1011

(a) Example graph

1

1 1 1 1 1 1

1 113 5 7 8 11

1

1111110 0 0 0 0

10 0

0 0 0 0 0

1 1 1 1 10 00 0 0 0

0 0 0 010642 9

(b) Four locally optimized partitionsin the population (subsets 0 and 1)

00

00

0

00

11

1

23

03

3

(c) Two detected clusters

Figure 2 An example of cluster detection

are moved to subset 119886 and cluster 119909 is removed from theset of candidate clusters This process is repeated until nocandidates remain or no move yields a positive cluster gainNo attempt is made to balance the partition during cluster-moving this is performed later

5 Memetic Search Framework

CMPA (cluster-moving memetic partitioning algorithm) is amemetic algorithm that we have designed for graph parti-tioning In this MA an individual is a 119896-way partition Eachgene in an individual corresponds to a vertex and has a valuebetween 0 and 119896 minus 1 which indicates the subset to which thevertex belongs that is the 119894th gene 119892

119894= 119895 hArr V

119894isin 119875119895+1

It isa steady-state MA meaning that there is only one offspringfrom population in each generation Crossover is followed bya cluster-moving step and then local optimization

Algorithm 2 shows the processes that make up CMPAwhich we now describe in detail

(i) Initialization When the MA starts 119898 individuals(ie partitions) are created at random Then eachindividual is improved by local optimization We set119898 to be 30 for bipartitioning and 50 for multiwaypartitioning (with 119896 = 8 or 119896 = 32)

6 Mathematical Problems in Engineering

Select clusters 119878 larr 119873 clusters selected at randomdo

Calculate cg(119909 119886) for all 119909 isin 119878 1 le 119886 le 119896cglowast larr max

119909119886cg(119909 119886)

(119909 119886) larr argmax119909119886

cg(119909 119886)

if cglowast gt 0 then

Move cluster 119909 to partition 119886119878 larr 119878 119909

until (119878 = 0 or cglowast le 0)

Algorithm 1 Our cluster-moving scheme

Create initial population of fixed sizeApply local optimization to each member of populationCalculate genic distance from populationFind clusters and store their informationdo

Select 1199011198861199031198901198991199051and 119901119886119903119890119899119905

2from population

Normalization(1199011198861199031198901198991199051 119901119886119903119890119899119905

2)

offspring larr Crossover(1199011198861199031198901198991199051 119901119886119903119890119899119905

2)

Cluster-moving(offspring)Local-optimization(offspring)Replace(population offspring)

until (stopping condition)return the best solution

Algorithm 2 The process in CMPA

(ii) Selection We used the roulette-wheel-based propor-tional selection The probability that the best individ-ual is chosen was set to four times the probabilitythat the worst is chosen The fitness value 119891

119894of the

119894th individual is expressed as (119888119908minus 119888119894) + (119888119908minus 119888119887)3

where 119888119887 119888119908 and 119888

119894are the cut sizes of the partitions

corresponding to the best the worst and the 119894thindividual respectively

(iii) Normalization Laszewski [31] first used normaliza-tion to improve the performance of GA and its vari-ants have been suggested in [23 26 41 42]The parentindividuals are normalized before crossover followingLaszewski [31 33] The subset of one parent whichshares the largest number of vertices with subsetsof the other parent is selected and that subset isnumbered 0 This process is repeated incrementingthe index until all subsets are numbered

(iv) Crossover and Cluster Moving We used a standardfive-point crossover After crossover the cluster-han-dling heuristic described in Section 42 is applied tothe individual At this point individuals usually cor-respond to unbalanced partitionsWe select a randomlocation in the individual and adjust the values ofthe genes which are the subsets to which the corre-sponding vertices belong to the right of this location(in a typical circular string) until the partition is

balanced This is effectively a mutation effect and nofurther mutation was introduced

(v) Local Optimization KL [7] was used for the bisectionproblems and EFM (extended FM) [21] was used formultiway partitioning (119896 = 8 or 119896 = 32)

(vi) Replacement We used a replacement scheme due toBui and Moon [10] If an offspring is better than itscloser parent the MA replaces that parent If it is bet-ter than its other parent then that parent is replacedOtherwise it replaces the worst individual in thepopulation

(vii) Stopping Condition This is based on consecutive fail-ures to replace an individualrsquos parents Termination istriggered by consecutive failures 30 in bipartitioningand 50 in multiway partitioning

6 Experimental Results

We conducted experiments on 2-way 8-way and 32-way par-titioning Table 4 shows the performance of MA combinedwith KL (denoted KL-MA) and CMPA on bipartitioningTable 5 shows the performance of the genetic extended FMalgorithm (GEFM) [21] one of themost effective approachesand CMPA on 8-way partitioning and Table 6 gives theresults for 32-way partitioning CMPAuses a cluster-handling

Mathematical Problems in Engineering 7

Table 4 Comparison of KL-MA and CMPA on bipartitioning

Graph Best known1 KL-MA CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

G50025 49 49 5134 032 (595) 49 5122 034 (558)G50005 218 218 21865 059 (700) 218 21845 064 (682)G50010 626 626 62755 102 (681) 626 62746 110 (672)G50020 1744 1744 174583 193 (694) 1744 174562 209 (687)G100025 93 93 9726 096 (745) 93 9708 104 (722)G100005 445 445 45150 207 (884) 445 45092 222 (860)G100010 1362 1362 136668 407 (969) 1362 136618 445 (977)G100020 3382 3382 338561 742 (971) 3382 338526 813 (987)U50005 2 2 496 055 (793) 2 394 066 (803)U50010 26 26 2631 045 (459) 26 2604 057 (465)U50020 178 178 17806 042 (300) 178 17800 053 (271)U50040 412 412 41200 034 (190) 412 41200 037 (102)U100005 1 1 1167 171 (1077) 1 881 212 (1068)U100010 39 39 4627 151 (645) 39 4415 195 (653)U100020 222 222 22219 113 (347) 222 22207 151 (327)U100040 737 737 73700 120 (243) 737 73700 177 (205)1Best result from the literature2Best and average results from 1000 runs3CPU seconds on a Pentium IV 28GHz4Average number of generations over 1000 runs

Table 5 Comparison of GEFM and CMPA on 8-way partitioning

Graph GEFM [21] CMPABest1 Ave1 CPU2 (Gen3) Best1 Ave1 CPU2 (Gen3)

G50025 111 11521 4203 (2090) 111 11451 4025 (2091)G50005 465 46816 8565 (2314) 465 46763 7296 (2239)G50010 1254 125953 14454 (2347) 1254 125847 13504 (2297)G50020 3348 335480 29318 (2437) 3348 335371 28509 (2317)G100025 212 21630 14676 (3043) 211 21657 14842 (3023)G100005 930 93811 24720 (3148) 931 93926 23467 (3069)G100010 2714 272633 40896 (3146) 2711 272552 41698 (3083)G100020 6525 653655 79129 (3303) 6520 653835 78167 (3141)U50005 16 1880 7087 (2629) 16 1722 5589 (2137)U50010 143 14512 9875 (2277) 143 14426 9050 (1984)U50020 612 61457 12122 (1690) 611 61293 9753 (1483)U50040 1867 187260 15794 (1396) 1867 187173 12962 (1204)U100005 21 3366 18035 (3638) 20 2853 20953 (3172)U100010 176 18370 24308 (3105) 176 18230 25702 (2857)U100020 812 81405 23108 (1824) 812 81300 23293 (1756)U100040 2562 256305 24946 (1353) 2562 256285 27351 (1238)1Best and average results from 100 runs2CPU seconds on a Pentium IV 28GHz3Average number of generations over 100 runs

heuristic but KL-MA andGEFM do notWe performed 1000runs for bipartitioning and 100 runs for 8-way and 32-waypartitioning All the programs were written in the C languageand compiled using GNUrsquos gcc compiler It was run on aPentium IV 28GHz computer with Linux In the tables thebold-faced numbers indicate the lower of the average cut sizes

obtained by the two algorithms CMPA outperformed KL-MA and GEFM on most graphs with comparable runningtimes

CMPA underperformed on some random graphs whichmay be due to the weak cluster structures in these graphsCMPArsquos average performance was better on all the geometric

8 Mathematical Problems in Engineering

Table 6 Comparison of GEFM and CMPA on 32-way partitioning

Graph Best known1 GEFM [21] CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

G50025 177 177 18169 7898 (1491) 177 18082 7430 (1382)G50005 623 624 63047 16226 (2197) 623 62992 15918 (2035)G50010 1573 1574 158194 30083 (2364) 1572lowast 157972 35573 (2332)G50020 4034 4037 404506 68153 (2565) 4035 404281 67319 (2479)G100025 312 313 32099 33579 (2798) 313 32025 35644 (2648)G100005 1217 1208 121872 69867 (3133) 1205lowast 121689 73806 (3203)G100010 3360 3353lowast 336731 117148 (3294) 3355 336608 132359 (3267)G100020 7818 7817 782981 201278 (3475) 7815lowast 783048 236369 (3342)U50005 109 112 11639 13829 (1640) 109 11318 12115 (1595)U50010 523 531 53775 22552 (1512) 526 53184 16626 (1222)U50020 1825 1831 184139 28507 (1344) 1823lowast 183188 29947 (1197)U50040 5328 5364 538001 56174 (1381) 5348 536983 52315 (1331)U100005 117 118 12602 45103 (3137) 115lowast 12349 46435 (2967)U100010 577 576 58326 59901 (2599) 571lowast 57816 74350 (2603)U100020 2367 2375 239633 79877 (1963) 2373 238893 93043 (1848)U100040 7329 7399 741749 142178 (1634) 7382 740718 149310 (1573)1Best known values [20ndash22 24 33]2Best and average results from 100 runs Asterisked numbers are new best values3CPU seconds on a Pentium IV 28GHz4Average number of generations from 100 runs

graphs This suggests that effective cluster handling is moreimportant on the geometric graphs as we suggested inSection 31

The local optimization algorithm ismuchmore expensivethan the cluster-handling heuristic thus CMPA does nottake much longer to run than KL-MA or GEFM On averageCMPA required 1414more time than KL-MA for the bipar-titioning problems and 202 more than GEFM in 32-waypartitioning CMPA ran 552 faster in 8-way partitioning

7 Concluding Remarks

We devised a cluster-moving heuristic and combined it witha memetic algorithm (MA) for graph partitioning Experi-ments on 2-way 8-way and 32-way partitioning showed thatthis heuristic significantly improved the performance of theMA especially on the 32-way partitioning

The method of moving clusters that we have introducedaddresses a significant weakness in standard IIP algorithmsThe idea of adding an operation to complement a localoptimization algorithm might be used to address otherdeficiencies in MAs

Our method of cluster detection is based on a measurethat we call genic distance which is designed to reflect theextent to which two vertices connected by an edge belongto the same cluster Instead of computing genic distancesonce in an initialization phase anMA could recompute thesedistances as it progresses this might improve the accuracyof cluster detection at some cost in time We believe thatgenic distance might also be useful in solving other problemsinvolving clusters such as the maximum clique problem

Table 7 Real-world benchmark graphs

Graph Number of vertices Number of edgesnasa4704 4704 50026bcspwr09 1723 2394bcsstk13 2003 40940DEBR12 4096 8189

Appendix

Results on Real-World Graphs

We also tested on four real-world benchmark graphs usedin [11 15 43] The sizes of the graphs are given in Table 7We conducted experiments on 2-way and 8-way partitioningWe performed 100 runs for bipartitioning and 50 runs for8-way partitioning It was run on an Intel Core i7 36GHzcomputer with Linux Table 8 compares CMPA with KL-MAon bipartitioning and Table 9 compares CMPA with GEFMon 8-way partitioning In the tables the bold-faced numbersindicate the lower of the average cut sizes obtained by the twoalgorithms Similarly to the results in Section 6CMPAoverallperformed better than the others with comparable runningtimes

Disclosure

Apreliminary version of this paper appeared inProceedings ofthe Genetic and Evolutionary Computation Conference 2007(p 1520)

Mathematical Problems in Engineering 9

Table 8 Comparison of KL-MA and CMPA on bipartitioning

Graph Best known1 KL-MA CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

nasa4704 1292 1292 129200 415 (407) 1292 129200 382 (347)bcspwr09 9 9 1131 061 (1010) 9 1060 081 (1003)bcsstk13 2355 2355 235500 117 (330) 2355 235500 146 (282)DEBR12 548 548 54844 991 (765) 548 54820 1032 (794)1Best result from the literature2Best and average results from 100 runs3CPU seconds on Intel Core i7 36GHz4Average number of generations over 100 runs

Table 9 Comparison of GEFM and CMPA on 8-way partitioning

Graph GEFM [21] CMPABest1 Ave1 CPU2 (Gen3) Best1 Ave1 CPU2 (Gen3)

nasa4704 3898 390366 57569 (2235) 3896 390266 59151 (2255)bcspwr09 53 5708 7880 (2947) 54 5774 8516 (2925)bcsstk13 8911 893910 28119 (1427) 8919 893614 30154 (1476)DEBR12 1248 126016 31597 (3969) 1248 125990 35383 (4002)1Best and average results from 50 runs2CPU seconds on Intel Core i7 36GHz3Average number of generations over 50 runs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the ICT RampD program ofMSIPIITP Korea (10045253 The development of SDNOpenFlow-Based Enterprise Network Controller Technol-ogy) project The authors would like to thank ProfessorByung-Ro Moon and Dr Yongjoo Song for their valuablesuggestions for improving this paper The authors also thankJong-Pil Kim for providing the source code of GEFM [21]

References

[1] Y Kanizo D Hay and I Keslassy ldquoPalette distributing tablesin software-defined networksrdquo in Proceedings of the IEEEInternational Conference on Computer Communications pp545ndash549 April 2013

[2] P Wette M Draxler and A Schwabe ldquoMaxiNet distributedemulation of software-defined networksrdquo in Proceedings of theIFIP Networking Conference pp 1ndash9 June 2014

[3] M R Garey and D S Johnson Computers and Intractability AGuide to the Theory of NP-Completeness W H Freeman 1979

[4] T N Bui and C Jones ldquoFinding good approximate vertex andedge partitions is NP-hardrdquo Information Processing Letters vol42 no 3 pp 153ndash159 1992

[5] Y-H Kim ldquoImproved implementation choices for iterativeimprovement partitioning algorithms on circuitsrdquo in Proceed-ings of the International Conference on Computer Design pp 30ndash34 July 2008

[6] Y Yoon and Y-H Kim ldquoNew bucket managements in iterativeimprovement partitioning algorithmsrdquo Applied Mathematics ampInformation Sciences vol 7 no 2 pp 529ndash532 2013

[7] BW Kernighan and S Lin ldquoAn efficient heuristic procedure forpartitioning graphsrdquo Bell System Technical Journal vol 49 no2 pp 291ndash307 1970

[8] C Fiduccia and R Mattheyses ldquoA Linear-time heuristic forimproving network partitionsrdquo in Proceedings of the Conferenceon Design Automation pp 175ndash181 June 1982

[9] D S Johnson C R Aragon L A McGeoch and C SchevonldquoOptimization by simulated annealing An experimental evalu-ation Part I Graph partitioningrdquo Operations Research vol 37no 6 pp 865ndash892 1989

[10] T N Bui and B R Moon ldquoGenetic algorithm and graphpartitioningrdquo IEEETransactions on Computers vol 45 no 7 pp841ndash855 1996

[11] R Battiti and A A Bertossi ldquoGreedy prohibition and reactiveheuristics for graph partitioningrdquo IEEE Transactions on Com-puters vol 48 no 4 pp 361ndash385 1999

[12] E Rolland H Pirkul and F Glover ldquoTabu search for graphpartitioningrdquo Annals of Operations Research vol 63 no 2 pp209ndash232 1996

[13] T N Bui and L C Strite ldquoAn ant system algorithm for graphbisectionrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference pp 43ndash51 July 2002

[14] I Hwang Y-H Kim and B-R Moon ldquoMulti-attractor genereordering for graph bisectionrdquo inProceedings of theGenetic andEvolutionary Computation Conference pp 1209ndash1215 July 2006

[15] Y-H Kim and B-R Moon ldquoLock gain based graph partition-ingrdquo Journal of Heuristics vol 10 no 1 pp 37ndash57 2004

[16] P Merz and B Freisleben ldquoFitness landscapes memetic algo-rithms and greedy operators for graph bipartitioningrdquo Evolu-tionary Computation vol 8 no 1 pp 61ndash91 2000

[17] HMuhlenbein and TMahnig ldquoEvolutionary optimization andthe estimation of search distributions with applications to graph

10 Mathematical Problems in Engineering

bipartitioningrdquo International Journal of Approximate Reasoningvol 31 no 3 pp 157ndash192 2002

[18] A G Steenbeek E Marchiori and A E Eiben ldquoFindingbalanced graph bi-partitions using a hybrid genetic algorithmrdquoin Proceedings of the IEEE International Conference on Evolu-tionary Computation pp 90ndash95 May 1998

[19] Y Yoon and Y-H Kim ldquoVertex ordering clustering andtheir application to graph partitioningrdquoApplied Mathematics ampInformation Sciences vol 8 no 1 pp 135ndash138 2014

[20] S-J Kang and B-R Moon ldquoA hybrid genetic algorithm formultiway graph partitioningrdquo in Proceedings of the Genetic andEvolutionary Computation Conference pp 159ndash166 July 2000

[21] J-P Kim and B-R Moon ldquoA hybrid genetic search for multi-way graph partitioning based on direct partitioningrdquo in Pro-ceedings of the Genetic and Evolutionary Computation Confer-ence pp 408ndash415 July 2001

[22] Y-H Kim A Moraglio Y Yoon and B-R Moon ldquoGeometriccrossover for multiway graph partitioningrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1217ndash1224 July 2006

[23] A Moraglio Y-H Kim Y Yoon and B-R Moon ldquoGeometriccrossovers for multiway graph partitioningrdquo Evolutionary Com-putation vol 15 no 4 pp 445ndash474 2007

[24] AMoraglio Y-H Kim Y Yoon B-RMoon and R Poli ldquoGeo-metric crossover for permutations with repetitions applicationto graph partitioningrdquo in Proceedings of the PPSN Workshopon Evolutionary AlgorithmsmdashBridging Theory and PracticeSeptember 2006

[25] A J Soper C Walshaw and M Cross ldquoA combined evolu-tionary search and multilevel optimisation approach to graph-partitioningrdquo Journal of Global Optimization vol 29 no 2 pp225ndash241 2004

[26] Y Yoon Y-H Kim A Moraglio and B-R Moon ldquoQuotientgeometric crossovers and redundant encodingsrdquo TheoreticalComputer Science vol 425 pp 4ndash16 2012

[27] J Kim I Hwang Y-H Kim and B-R Moon ldquoGeneticapproaches for graph partitioning a surveyrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 473ndash480 July 2011

[28] S Dutt and W Deng ldquoA probability-based approach to VLSIcircuit partitioningrdquo in Proceedings of the Design AutomationConference pp 100ndash105 June 1996

[29] B Krishnamurthy ldquoAn improved min-cut algorithm for parti-tioning VLSI networksrdquo IEEE Transactions on Computers vol33 no 5 pp 438ndash446 1984

[30] L A Sanchis ldquoMultiple-way network partitioningrdquo IEEETrans-actions on Computers vol 38 no 1 pp 62ndash81 1989

[31] G Laszewski ldquoIntelligent structural operators for the k-waygraph partitioning problemrdquo in Proceedings of the InternationalConference on Genetic Algorithms pp 45ndash52 July 1991

[32] H Muhlenbein ldquoParallel genetic algorithms in combinatorialoptimizationrdquo in Computer Science and Operations Research OBalci R Sharda and S Zenios Eds pp 441ndash453 PergamonOxford UK 1992

[33] S-S Choi and B-R Moon ldquoNormalization in genetic algo-rithmsrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference vol 2723 of Lecture Notes in ComputerScience pp 862ndash873 2003

[34] S-H Kim Y-H Kim and B-R Moon ldquoA hybrid geneticalgorithm for themax cut problemrdquo inProceedings of theGeneticand Evolutionary Computation Conference pp 416ndash423 July2001

[35] Y-H Kim Y-K Kwon and B-R Moon ldquoProblem-independ-ent schema synthesis for genetic algorithmsrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1112ndash1122 July 2003

[36] K Seo S Hyun and Y-H Kim ldquoAn edge-set representationbased on spanning tree for searching cut spacerdquo IEEE Trans-actions on Evolutionary Computation 2014

[37] Y-H Kim Combinatorial optimization based on effective localsearch genetic algorithms and problem space analyses [PhDthesis] Seoul National University Seoul Republic of Korea2005

[38] Y-H Kim ldquoAn enzyme-inspired approach to surmount barriersin graph bisectionrdquo in Proceedings of the International Confer-ence on Computational Science and Its Applications vol 5072 ofLecture Notes in Computer Science pp 841ndash851 2008

[39] S Dutt and W Deng ldquoVLSI circuit partitioning by cluster-removal using iterative improvement techniquesrdquo in Proceed-ings of the IEEEACM International Conference on Computer-Aided Design pp 194ndash200 November 1996

[40] S Dutt and W Deng ldquoCluster-aware iterative improvementtechniques for partitioning large VLSI circuitsrdquo ACM Transac-tions on Design Automation of Electronic Systems vol 7 no 1pp 91ndash121 2002

[41] Y Yoon and Y-H Kim ldquoAn efficient genetic algorithm formaximum coverage deployment in wireless sensor networksrdquoIEEE Transactions on Cybernetics vol 43 no 5 pp 1473ndash14832013

[42] Y Yoon Y-H Kim AMoraglio and B-RMoon ldquoA theoreticaland empirical study on unbiased boundary-extended crossoverfor real-valued representationrdquo Information Sciences vol 183no 1 pp 48ndash65 2012

[43] BMonien andRDiekmann ldquoA local graph partitioning heuris-tic meeting bisection boundsrdquo in Proceedings of the SIAM Con-ference on Parallel Processing for Scientific Computing March1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Therefore we have

119889119892(V119894 V119895) le 119889119892(V119894 V119896) + 119889119892(V119896 V119895) (8)

That is 119889119892satisfies the triangle inequality

Proposition 1 suggests that the measure 119889119892is reasonable

A pseudometric space is a generalization of a metric space inwhich points need not be distinguishable thus it is possiblethat 119889

119892(V119894 V119895) = 0 for some edge V

119894 V119895 with distinct vertices

V119894= V119895

Our heuristic detects clusters by collecting a number oflocal optima and computes genic distances for all the edges ineach graphThis takes119874(119898|119864|) time but this cost is negligiblesince this computation is a preprocess performed before theMA runs The heuristic temporarily eliminates edges withgenic distances that are greater than a threshold value 120579 Weset 120579 to be the smallest value that satisfies

10038161003816100381610038161003816119890 isin 119864 119889

119892(119890) le 120579

10038161003816100381610038161003816le 01 |119881| (9)

Each remaining connected component containingmore thanthree vertices is considered to be a cluster

Figure 2 shows how our heuristic detects clusters Fig-ure 2(a) shows an example graph with 11 vertices and 15edges Four individuals corresponding to locally optimizedpartitions from the population are shown in Figure 2(b)In Figure 2(c) each edge is labeled with its genic distanceIf the threshold value of genic distance is 1 then theedges with larger genic distances indicated by dotted lineare eliminated Then four connected components remainV3 V7 V8 V9 V2 V4 V6 V11 V1 V5 and V

10 The last two

of these connected components are considered too small tobe clusters Thus clusters V

3 V7 V8 V9 and V

2 V4 V6 V11

shaded in Figure 2(c) remain as candidates for moving

42 Cluster-Moving Scheme Our heuristic improves theoffspring of each generation after crossover by moving theclusters that were detected using the technique described inthe previous subsection To select the clusters to be movedand their target subsets we introduce ameasure called clustergain such that cg(119909 119886) is the reduction in the cut size of thepartition when all the vertices in cluster 119909 are moved to thesubset 119886 For example moving the cluster in Figure 1(a) to theother subset in the partition is associated with a cluster gainof 2

This cluster-moving scheme described in Algorithm 1is applied to each individual generated by crossover whichis a partition that may be unbalanced However cut sizeand cluster gain are well defined on unbalanced partitionsOur scheme does not consider moving every cluster in everypartition because we found that making all clusters movablecauses the premature convergence of the MA Thus ourheuristic selects119873 clusters at random as candidates for mov-ing In our experiments we set119873 to 5We compute the clustergain that results from moving each candidate cluster to eachof the other 119896 minus 1 subsets The candidate cluster 119909 anddestination subset 119886with the highest cluster gain are selectedAssume that cg(119909 119886) is positive all the vertices in cluster 119909

6

12

3 45

78

9 1011

(a) Example graph

1

1 1 1 1 1 1

1 113 5 7 8 11

1

1111110 0 0 0 0

10 0

0 0 0 0 0

1 1 1 1 10 00 0 0 0

0 0 0 010642 9

(b) Four locally optimized partitionsin the population (subsets 0 and 1)

00

00

0

00

11

1

23

03

3

(c) Two detected clusters

Figure 2 An example of cluster detection

are moved to subset 119886 and cluster 119909 is removed from theset of candidate clusters This process is repeated until nocandidates remain or no move yields a positive cluster gainNo attempt is made to balance the partition during cluster-moving this is performed later

5 Memetic Search Framework

CMPA (cluster-moving memetic partitioning algorithm) is amemetic algorithm that we have designed for graph parti-tioning In this MA an individual is a 119896-way partition Eachgene in an individual corresponds to a vertex and has a valuebetween 0 and 119896 minus 1 which indicates the subset to which thevertex belongs that is the 119894th gene 119892

119894= 119895 hArr V

119894isin 119875119895+1

It isa steady-state MA meaning that there is only one offspringfrom population in each generation Crossover is followed bya cluster-moving step and then local optimization

Algorithm 2 shows the processes that make up CMPAwhich we now describe in detail

(i) Initialization When the MA starts 119898 individuals(ie partitions) are created at random Then eachindividual is improved by local optimization We set119898 to be 30 for bipartitioning and 50 for multiwaypartitioning (with 119896 = 8 or 119896 = 32)

6 Mathematical Problems in Engineering

Select clusters 119878 larr 119873 clusters selected at randomdo

Calculate cg(119909 119886) for all 119909 isin 119878 1 le 119886 le 119896cglowast larr max

119909119886cg(119909 119886)

(119909 119886) larr argmax119909119886

cg(119909 119886)

if cglowast gt 0 then

Move cluster 119909 to partition 119886119878 larr 119878 119909

until (119878 = 0 or cglowast le 0)

Algorithm 1 Our cluster-moving scheme

Create initial population of fixed sizeApply local optimization to each member of populationCalculate genic distance from populationFind clusters and store their informationdo

Select 1199011198861199031198901198991199051and 119901119886119903119890119899119905

2from population

Normalization(1199011198861199031198901198991199051 119901119886119903119890119899119905

2)

offspring larr Crossover(1199011198861199031198901198991199051 119901119886119903119890119899119905

2)

Cluster-moving(offspring)Local-optimization(offspring)Replace(population offspring)

until (stopping condition)return the best solution

Algorithm 2 The process in CMPA

(ii) Selection We used the roulette-wheel-based propor-tional selection The probability that the best individ-ual is chosen was set to four times the probabilitythat the worst is chosen The fitness value 119891

119894of the

119894th individual is expressed as (119888119908minus 119888119894) + (119888119908minus 119888119887)3

where 119888119887 119888119908 and 119888

119894are the cut sizes of the partitions

corresponding to the best the worst and the 119894thindividual respectively

(iii) Normalization Laszewski [31] first used normaliza-tion to improve the performance of GA and its vari-ants have been suggested in [23 26 41 42]The parentindividuals are normalized before crossover followingLaszewski [31 33] The subset of one parent whichshares the largest number of vertices with subsetsof the other parent is selected and that subset isnumbered 0 This process is repeated incrementingthe index until all subsets are numbered

(iv) Crossover and Cluster Moving We used a standardfive-point crossover After crossover the cluster-han-dling heuristic described in Section 42 is applied tothe individual At this point individuals usually cor-respond to unbalanced partitionsWe select a randomlocation in the individual and adjust the values ofthe genes which are the subsets to which the corre-sponding vertices belong to the right of this location(in a typical circular string) until the partition is

balanced This is effectively a mutation effect and nofurther mutation was introduced

(v) Local Optimization KL [7] was used for the bisectionproblems and EFM (extended FM) [21] was used formultiway partitioning (119896 = 8 or 119896 = 32)

(vi) Replacement We used a replacement scheme due toBui and Moon [10] If an offspring is better than itscloser parent the MA replaces that parent If it is bet-ter than its other parent then that parent is replacedOtherwise it replaces the worst individual in thepopulation

(vii) Stopping Condition This is based on consecutive fail-ures to replace an individualrsquos parents Termination istriggered by consecutive failures 30 in bipartitioningand 50 in multiway partitioning

6 Experimental Results

We conducted experiments on 2-way 8-way and 32-way par-titioning Table 4 shows the performance of MA combinedwith KL (denoted KL-MA) and CMPA on bipartitioningTable 5 shows the performance of the genetic extended FMalgorithm (GEFM) [21] one of themost effective approachesand CMPA on 8-way partitioning and Table 6 gives theresults for 32-way partitioning CMPAuses a cluster-handling

Mathematical Problems in Engineering 7

Table 4 Comparison of KL-MA and CMPA on bipartitioning

Graph Best known1 KL-MA CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

G50025 49 49 5134 032 (595) 49 5122 034 (558)G50005 218 218 21865 059 (700) 218 21845 064 (682)G50010 626 626 62755 102 (681) 626 62746 110 (672)G50020 1744 1744 174583 193 (694) 1744 174562 209 (687)G100025 93 93 9726 096 (745) 93 9708 104 (722)G100005 445 445 45150 207 (884) 445 45092 222 (860)G100010 1362 1362 136668 407 (969) 1362 136618 445 (977)G100020 3382 3382 338561 742 (971) 3382 338526 813 (987)U50005 2 2 496 055 (793) 2 394 066 (803)U50010 26 26 2631 045 (459) 26 2604 057 (465)U50020 178 178 17806 042 (300) 178 17800 053 (271)U50040 412 412 41200 034 (190) 412 41200 037 (102)U100005 1 1 1167 171 (1077) 1 881 212 (1068)U100010 39 39 4627 151 (645) 39 4415 195 (653)U100020 222 222 22219 113 (347) 222 22207 151 (327)U100040 737 737 73700 120 (243) 737 73700 177 (205)1Best result from the literature2Best and average results from 1000 runs3CPU seconds on a Pentium IV 28GHz4Average number of generations over 1000 runs

Table 5 Comparison of GEFM and CMPA on 8-way partitioning

Graph GEFM [21] CMPABest1 Ave1 CPU2 (Gen3) Best1 Ave1 CPU2 (Gen3)

G50025 111 11521 4203 (2090) 111 11451 4025 (2091)G50005 465 46816 8565 (2314) 465 46763 7296 (2239)G50010 1254 125953 14454 (2347) 1254 125847 13504 (2297)G50020 3348 335480 29318 (2437) 3348 335371 28509 (2317)G100025 212 21630 14676 (3043) 211 21657 14842 (3023)G100005 930 93811 24720 (3148) 931 93926 23467 (3069)G100010 2714 272633 40896 (3146) 2711 272552 41698 (3083)G100020 6525 653655 79129 (3303) 6520 653835 78167 (3141)U50005 16 1880 7087 (2629) 16 1722 5589 (2137)U50010 143 14512 9875 (2277) 143 14426 9050 (1984)U50020 612 61457 12122 (1690) 611 61293 9753 (1483)U50040 1867 187260 15794 (1396) 1867 187173 12962 (1204)U100005 21 3366 18035 (3638) 20 2853 20953 (3172)U100010 176 18370 24308 (3105) 176 18230 25702 (2857)U100020 812 81405 23108 (1824) 812 81300 23293 (1756)U100040 2562 256305 24946 (1353) 2562 256285 27351 (1238)1Best and average results from 100 runs2CPU seconds on a Pentium IV 28GHz3Average number of generations over 100 runs

heuristic but KL-MA andGEFM do notWe performed 1000runs for bipartitioning and 100 runs for 8-way and 32-waypartitioning All the programs were written in the C languageand compiled using GNUrsquos gcc compiler It was run on aPentium IV 28GHz computer with Linux In the tables thebold-faced numbers indicate the lower of the average cut sizes

obtained by the two algorithms CMPA outperformed KL-MA and GEFM on most graphs with comparable runningtimes

CMPA underperformed on some random graphs whichmay be due to the weak cluster structures in these graphsCMPArsquos average performance was better on all the geometric

8 Mathematical Problems in Engineering

Table 6 Comparison of GEFM and CMPA on 32-way partitioning

Graph Best known1 GEFM [21] CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

G50025 177 177 18169 7898 (1491) 177 18082 7430 (1382)G50005 623 624 63047 16226 (2197) 623 62992 15918 (2035)G50010 1573 1574 158194 30083 (2364) 1572lowast 157972 35573 (2332)G50020 4034 4037 404506 68153 (2565) 4035 404281 67319 (2479)G100025 312 313 32099 33579 (2798) 313 32025 35644 (2648)G100005 1217 1208 121872 69867 (3133) 1205lowast 121689 73806 (3203)G100010 3360 3353lowast 336731 117148 (3294) 3355 336608 132359 (3267)G100020 7818 7817 782981 201278 (3475) 7815lowast 783048 236369 (3342)U50005 109 112 11639 13829 (1640) 109 11318 12115 (1595)U50010 523 531 53775 22552 (1512) 526 53184 16626 (1222)U50020 1825 1831 184139 28507 (1344) 1823lowast 183188 29947 (1197)U50040 5328 5364 538001 56174 (1381) 5348 536983 52315 (1331)U100005 117 118 12602 45103 (3137) 115lowast 12349 46435 (2967)U100010 577 576 58326 59901 (2599) 571lowast 57816 74350 (2603)U100020 2367 2375 239633 79877 (1963) 2373 238893 93043 (1848)U100040 7329 7399 741749 142178 (1634) 7382 740718 149310 (1573)1Best known values [20ndash22 24 33]2Best and average results from 100 runs Asterisked numbers are new best values3CPU seconds on a Pentium IV 28GHz4Average number of generations from 100 runs

graphs This suggests that effective cluster handling is moreimportant on the geometric graphs as we suggested inSection 31

The local optimization algorithm ismuchmore expensivethan the cluster-handling heuristic thus CMPA does nottake much longer to run than KL-MA or GEFM On averageCMPA required 1414more time than KL-MA for the bipar-titioning problems and 202 more than GEFM in 32-waypartitioning CMPA ran 552 faster in 8-way partitioning

7 Concluding Remarks

We devised a cluster-moving heuristic and combined it witha memetic algorithm (MA) for graph partitioning Experi-ments on 2-way 8-way and 32-way partitioning showed thatthis heuristic significantly improved the performance of theMA especially on the 32-way partitioning

The method of moving clusters that we have introducedaddresses a significant weakness in standard IIP algorithmsThe idea of adding an operation to complement a localoptimization algorithm might be used to address otherdeficiencies in MAs

Our method of cluster detection is based on a measurethat we call genic distance which is designed to reflect theextent to which two vertices connected by an edge belongto the same cluster Instead of computing genic distancesonce in an initialization phase anMA could recompute thesedistances as it progresses this might improve the accuracyof cluster detection at some cost in time We believe thatgenic distance might also be useful in solving other problemsinvolving clusters such as the maximum clique problem

Table 7 Real-world benchmark graphs

Graph Number of vertices Number of edgesnasa4704 4704 50026bcspwr09 1723 2394bcsstk13 2003 40940DEBR12 4096 8189

Appendix

Results on Real-World Graphs

We also tested on four real-world benchmark graphs usedin [11 15 43] The sizes of the graphs are given in Table 7We conducted experiments on 2-way and 8-way partitioningWe performed 100 runs for bipartitioning and 50 runs for8-way partitioning It was run on an Intel Core i7 36GHzcomputer with Linux Table 8 compares CMPA with KL-MAon bipartitioning and Table 9 compares CMPA with GEFMon 8-way partitioning In the tables the bold-faced numbersindicate the lower of the average cut sizes obtained by the twoalgorithms Similarly to the results in Section 6CMPAoverallperformed better than the others with comparable runningtimes

Disclosure

Apreliminary version of this paper appeared inProceedings ofthe Genetic and Evolutionary Computation Conference 2007(p 1520)

Mathematical Problems in Engineering 9

Table 8 Comparison of KL-MA and CMPA on bipartitioning

Graph Best known1 KL-MA CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

nasa4704 1292 1292 129200 415 (407) 1292 129200 382 (347)bcspwr09 9 9 1131 061 (1010) 9 1060 081 (1003)bcsstk13 2355 2355 235500 117 (330) 2355 235500 146 (282)DEBR12 548 548 54844 991 (765) 548 54820 1032 (794)1Best result from the literature2Best and average results from 100 runs3CPU seconds on Intel Core i7 36GHz4Average number of generations over 100 runs

Table 9 Comparison of GEFM and CMPA on 8-way partitioning

Graph GEFM [21] CMPABest1 Ave1 CPU2 (Gen3) Best1 Ave1 CPU2 (Gen3)

nasa4704 3898 390366 57569 (2235) 3896 390266 59151 (2255)bcspwr09 53 5708 7880 (2947) 54 5774 8516 (2925)bcsstk13 8911 893910 28119 (1427) 8919 893614 30154 (1476)DEBR12 1248 126016 31597 (3969) 1248 125990 35383 (4002)1Best and average results from 50 runs2CPU seconds on Intel Core i7 36GHz3Average number of generations over 50 runs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the ICT RampD program ofMSIPIITP Korea (10045253 The development of SDNOpenFlow-Based Enterprise Network Controller Technol-ogy) project The authors would like to thank ProfessorByung-Ro Moon and Dr Yongjoo Song for their valuablesuggestions for improving this paper The authors also thankJong-Pil Kim for providing the source code of GEFM [21]

References

[1] Y Kanizo D Hay and I Keslassy ldquoPalette distributing tablesin software-defined networksrdquo in Proceedings of the IEEEInternational Conference on Computer Communications pp545ndash549 April 2013

[2] P Wette M Draxler and A Schwabe ldquoMaxiNet distributedemulation of software-defined networksrdquo in Proceedings of theIFIP Networking Conference pp 1ndash9 June 2014

[3] M R Garey and D S Johnson Computers and Intractability AGuide to the Theory of NP-Completeness W H Freeman 1979

[4] T N Bui and C Jones ldquoFinding good approximate vertex andedge partitions is NP-hardrdquo Information Processing Letters vol42 no 3 pp 153ndash159 1992

[5] Y-H Kim ldquoImproved implementation choices for iterativeimprovement partitioning algorithms on circuitsrdquo in Proceed-ings of the International Conference on Computer Design pp 30ndash34 July 2008

[6] Y Yoon and Y-H Kim ldquoNew bucket managements in iterativeimprovement partitioning algorithmsrdquo Applied Mathematics ampInformation Sciences vol 7 no 2 pp 529ndash532 2013

[7] BW Kernighan and S Lin ldquoAn efficient heuristic procedure forpartitioning graphsrdquo Bell System Technical Journal vol 49 no2 pp 291ndash307 1970

[8] C Fiduccia and R Mattheyses ldquoA Linear-time heuristic forimproving network partitionsrdquo in Proceedings of the Conferenceon Design Automation pp 175ndash181 June 1982

[9] D S Johnson C R Aragon L A McGeoch and C SchevonldquoOptimization by simulated annealing An experimental evalu-ation Part I Graph partitioningrdquo Operations Research vol 37no 6 pp 865ndash892 1989

[10] T N Bui and B R Moon ldquoGenetic algorithm and graphpartitioningrdquo IEEETransactions on Computers vol 45 no 7 pp841ndash855 1996

[11] R Battiti and A A Bertossi ldquoGreedy prohibition and reactiveheuristics for graph partitioningrdquo IEEE Transactions on Com-puters vol 48 no 4 pp 361ndash385 1999

[12] E Rolland H Pirkul and F Glover ldquoTabu search for graphpartitioningrdquo Annals of Operations Research vol 63 no 2 pp209ndash232 1996

[13] T N Bui and L C Strite ldquoAn ant system algorithm for graphbisectionrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference pp 43ndash51 July 2002

[14] I Hwang Y-H Kim and B-R Moon ldquoMulti-attractor genereordering for graph bisectionrdquo inProceedings of theGenetic andEvolutionary Computation Conference pp 1209ndash1215 July 2006

[15] Y-H Kim and B-R Moon ldquoLock gain based graph partition-ingrdquo Journal of Heuristics vol 10 no 1 pp 37ndash57 2004

[16] P Merz and B Freisleben ldquoFitness landscapes memetic algo-rithms and greedy operators for graph bipartitioningrdquo Evolu-tionary Computation vol 8 no 1 pp 61ndash91 2000

[17] HMuhlenbein and TMahnig ldquoEvolutionary optimization andthe estimation of search distributions with applications to graph

10 Mathematical Problems in Engineering

bipartitioningrdquo International Journal of Approximate Reasoningvol 31 no 3 pp 157ndash192 2002

[18] A G Steenbeek E Marchiori and A E Eiben ldquoFindingbalanced graph bi-partitions using a hybrid genetic algorithmrdquoin Proceedings of the IEEE International Conference on Evolu-tionary Computation pp 90ndash95 May 1998

[19] Y Yoon and Y-H Kim ldquoVertex ordering clustering andtheir application to graph partitioningrdquoApplied Mathematics ampInformation Sciences vol 8 no 1 pp 135ndash138 2014

[20] S-J Kang and B-R Moon ldquoA hybrid genetic algorithm formultiway graph partitioningrdquo in Proceedings of the Genetic andEvolutionary Computation Conference pp 159ndash166 July 2000

[21] J-P Kim and B-R Moon ldquoA hybrid genetic search for multi-way graph partitioning based on direct partitioningrdquo in Pro-ceedings of the Genetic and Evolutionary Computation Confer-ence pp 408ndash415 July 2001

[22] Y-H Kim A Moraglio Y Yoon and B-R Moon ldquoGeometriccrossover for multiway graph partitioningrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1217ndash1224 July 2006

[23] A Moraglio Y-H Kim Y Yoon and B-R Moon ldquoGeometriccrossovers for multiway graph partitioningrdquo Evolutionary Com-putation vol 15 no 4 pp 445ndash474 2007

[24] AMoraglio Y-H Kim Y Yoon B-RMoon and R Poli ldquoGeo-metric crossover for permutations with repetitions applicationto graph partitioningrdquo in Proceedings of the PPSN Workshopon Evolutionary AlgorithmsmdashBridging Theory and PracticeSeptember 2006

[25] A J Soper C Walshaw and M Cross ldquoA combined evolu-tionary search and multilevel optimisation approach to graph-partitioningrdquo Journal of Global Optimization vol 29 no 2 pp225ndash241 2004

[26] Y Yoon Y-H Kim A Moraglio and B-R Moon ldquoQuotientgeometric crossovers and redundant encodingsrdquo TheoreticalComputer Science vol 425 pp 4ndash16 2012

[27] J Kim I Hwang Y-H Kim and B-R Moon ldquoGeneticapproaches for graph partitioning a surveyrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 473ndash480 July 2011

[28] S Dutt and W Deng ldquoA probability-based approach to VLSIcircuit partitioningrdquo in Proceedings of the Design AutomationConference pp 100ndash105 June 1996

[29] B Krishnamurthy ldquoAn improved min-cut algorithm for parti-tioning VLSI networksrdquo IEEE Transactions on Computers vol33 no 5 pp 438ndash446 1984

[30] L A Sanchis ldquoMultiple-way network partitioningrdquo IEEETrans-actions on Computers vol 38 no 1 pp 62ndash81 1989

[31] G Laszewski ldquoIntelligent structural operators for the k-waygraph partitioning problemrdquo in Proceedings of the InternationalConference on Genetic Algorithms pp 45ndash52 July 1991

[32] H Muhlenbein ldquoParallel genetic algorithms in combinatorialoptimizationrdquo in Computer Science and Operations Research OBalci R Sharda and S Zenios Eds pp 441ndash453 PergamonOxford UK 1992

[33] S-S Choi and B-R Moon ldquoNormalization in genetic algo-rithmsrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference vol 2723 of Lecture Notes in ComputerScience pp 862ndash873 2003

[34] S-H Kim Y-H Kim and B-R Moon ldquoA hybrid geneticalgorithm for themax cut problemrdquo inProceedings of theGeneticand Evolutionary Computation Conference pp 416ndash423 July2001

[35] Y-H Kim Y-K Kwon and B-R Moon ldquoProblem-independ-ent schema synthesis for genetic algorithmsrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1112ndash1122 July 2003

[36] K Seo S Hyun and Y-H Kim ldquoAn edge-set representationbased on spanning tree for searching cut spacerdquo IEEE Trans-actions on Evolutionary Computation 2014

[37] Y-H Kim Combinatorial optimization based on effective localsearch genetic algorithms and problem space analyses [PhDthesis] Seoul National University Seoul Republic of Korea2005

[38] Y-H Kim ldquoAn enzyme-inspired approach to surmount barriersin graph bisectionrdquo in Proceedings of the International Confer-ence on Computational Science and Its Applications vol 5072 ofLecture Notes in Computer Science pp 841ndash851 2008

[39] S Dutt and W Deng ldquoVLSI circuit partitioning by cluster-removal using iterative improvement techniquesrdquo in Proceed-ings of the IEEEACM International Conference on Computer-Aided Design pp 194ndash200 November 1996

[40] S Dutt and W Deng ldquoCluster-aware iterative improvementtechniques for partitioning large VLSI circuitsrdquo ACM Transac-tions on Design Automation of Electronic Systems vol 7 no 1pp 91ndash121 2002

[41] Y Yoon and Y-H Kim ldquoAn efficient genetic algorithm formaximum coverage deployment in wireless sensor networksrdquoIEEE Transactions on Cybernetics vol 43 no 5 pp 1473ndash14832013

[42] Y Yoon Y-H Kim AMoraglio and B-RMoon ldquoA theoreticaland empirical study on unbiased boundary-extended crossoverfor real-valued representationrdquo Information Sciences vol 183no 1 pp 48ndash65 2012

[43] BMonien andRDiekmann ldquoA local graph partitioning heuris-tic meeting bisection boundsrdquo in Proceedings of the SIAM Con-ference on Parallel Processing for Scientific Computing March1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Select clusters 119878 larr 119873 clusters selected at randomdo

Calculate cg(119909 119886) for all 119909 isin 119878 1 le 119886 le 119896cglowast larr max

119909119886cg(119909 119886)

(119909 119886) larr argmax119909119886

cg(119909 119886)

if cglowast gt 0 then

Move cluster 119909 to partition 119886119878 larr 119878 119909

until (119878 = 0 or cglowast le 0)

Algorithm 1 Our cluster-moving scheme

Create initial population of fixed sizeApply local optimization to each member of populationCalculate genic distance from populationFind clusters and store their informationdo

Select 1199011198861199031198901198991199051and 119901119886119903119890119899119905

2from population

Normalization(1199011198861199031198901198991199051 119901119886119903119890119899119905

2)

offspring larr Crossover(1199011198861199031198901198991199051 119901119886119903119890119899119905

2)

Cluster-moving(offspring)Local-optimization(offspring)Replace(population offspring)

until (stopping condition)return the best solution

Algorithm 2 The process in CMPA

(ii) Selection We used the roulette-wheel-based propor-tional selection The probability that the best individ-ual is chosen was set to four times the probabilitythat the worst is chosen The fitness value 119891

119894of the

119894th individual is expressed as (119888119908minus 119888119894) + (119888119908minus 119888119887)3

where 119888119887 119888119908 and 119888

119894are the cut sizes of the partitions

corresponding to the best the worst and the 119894thindividual respectively

(iii) Normalization Laszewski [31] first used normaliza-tion to improve the performance of GA and its vari-ants have been suggested in [23 26 41 42]The parentindividuals are normalized before crossover followingLaszewski [31 33] The subset of one parent whichshares the largest number of vertices with subsetsof the other parent is selected and that subset isnumbered 0 This process is repeated incrementingthe index until all subsets are numbered

(iv) Crossover and Cluster Moving We used a standardfive-point crossover After crossover the cluster-han-dling heuristic described in Section 42 is applied tothe individual At this point individuals usually cor-respond to unbalanced partitionsWe select a randomlocation in the individual and adjust the values ofthe genes which are the subsets to which the corre-sponding vertices belong to the right of this location(in a typical circular string) until the partition is

balanced This is effectively a mutation effect and nofurther mutation was introduced

(v) Local Optimization KL [7] was used for the bisectionproblems and EFM (extended FM) [21] was used formultiway partitioning (119896 = 8 or 119896 = 32)

(vi) Replacement We used a replacement scheme due toBui and Moon [10] If an offspring is better than itscloser parent the MA replaces that parent If it is bet-ter than its other parent then that parent is replacedOtherwise it replaces the worst individual in thepopulation

(vii) Stopping Condition This is based on consecutive fail-ures to replace an individualrsquos parents Termination istriggered by consecutive failures 30 in bipartitioningand 50 in multiway partitioning

6 Experimental Results

We conducted experiments on 2-way 8-way and 32-way par-titioning Table 4 shows the performance of MA combinedwith KL (denoted KL-MA) and CMPA on bipartitioningTable 5 shows the performance of the genetic extended FMalgorithm (GEFM) [21] one of themost effective approachesand CMPA on 8-way partitioning and Table 6 gives theresults for 32-way partitioning CMPAuses a cluster-handling

Mathematical Problems in Engineering 7

Table 4 Comparison of KL-MA and CMPA on bipartitioning

Graph Best known1 KL-MA CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

G50025 49 49 5134 032 (595) 49 5122 034 (558)G50005 218 218 21865 059 (700) 218 21845 064 (682)G50010 626 626 62755 102 (681) 626 62746 110 (672)G50020 1744 1744 174583 193 (694) 1744 174562 209 (687)G100025 93 93 9726 096 (745) 93 9708 104 (722)G100005 445 445 45150 207 (884) 445 45092 222 (860)G100010 1362 1362 136668 407 (969) 1362 136618 445 (977)G100020 3382 3382 338561 742 (971) 3382 338526 813 (987)U50005 2 2 496 055 (793) 2 394 066 (803)U50010 26 26 2631 045 (459) 26 2604 057 (465)U50020 178 178 17806 042 (300) 178 17800 053 (271)U50040 412 412 41200 034 (190) 412 41200 037 (102)U100005 1 1 1167 171 (1077) 1 881 212 (1068)U100010 39 39 4627 151 (645) 39 4415 195 (653)U100020 222 222 22219 113 (347) 222 22207 151 (327)U100040 737 737 73700 120 (243) 737 73700 177 (205)1Best result from the literature2Best and average results from 1000 runs3CPU seconds on a Pentium IV 28GHz4Average number of generations over 1000 runs

Table 5 Comparison of GEFM and CMPA on 8-way partitioning

Graph GEFM [21] CMPABest1 Ave1 CPU2 (Gen3) Best1 Ave1 CPU2 (Gen3)

G50025 111 11521 4203 (2090) 111 11451 4025 (2091)G50005 465 46816 8565 (2314) 465 46763 7296 (2239)G50010 1254 125953 14454 (2347) 1254 125847 13504 (2297)G50020 3348 335480 29318 (2437) 3348 335371 28509 (2317)G100025 212 21630 14676 (3043) 211 21657 14842 (3023)G100005 930 93811 24720 (3148) 931 93926 23467 (3069)G100010 2714 272633 40896 (3146) 2711 272552 41698 (3083)G100020 6525 653655 79129 (3303) 6520 653835 78167 (3141)U50005 16 1880 7087 (2629) 16 1722 5589 (2137)U50010 143 14512 9875 (2277) 143 14426 9050 (1984)U50020 612 61457 12122 (1690) 611 61293 9753 (1483)U50040 1867 187260 15794 (1396) 1867 187173 12962 (1204)U100005 21 3366 18035 (3638) 20 2853 20953 (3172)U100010 176 18370 24308 (3105) 176 18230 25702 (2857)U100020 812 81405 23108 (1824) 812 81300 23293 (1756)U100040 2562 256305 24946 (1353) 2562 256285 27351 (1238)1Best and average results from 100 runs2CPU seconds on a Pentium IV 28GHz3Average number of generations over 100 runs

heuristic but KL-MA andGEFM do notWe performed 1000runs for bipartitioning and 100 runs for 8-way and 32-waypartitioning All the programs were written in the C languageand compiled using GNUrsquos gcc compiler It was run on aPentium IV 28GHz computer with Linux In the tables thebold-faced numbers indicate the lower of the average cut sizes

obtained by the two algorithms CMPA outperformed KL-MA and GEFM on most graphs with comparable runningtimes

CMPA underperformed on some random graphs whichmay be due to the weak cluster structures in these graphsCMPArsquos average performance was better on all the geometric

8 Mathematical Problems in Engineering

Table 6 Comparison of GEFM and CMPA on 32-way partitioning

Graph Best known1 GEFM [21] CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

G50025 177 177 18169 7898 (1491) 177 18082 7430 (1382)G50005 623 624 63047 16226 (2197) 623 62992 15918 (2035)G50010 1573 1574 158194 30083 (2364) 1572lowast 157972 35573 (2332)G50020 4034 4037 404506 68153 (2565) 4035 404281 67319 (2479)G100025 312 313 32099 33579 (2798) 313 32025 35644 (2648)G100005 1217 1208 121872 69867 (3133) 1205lowast 121689 73806 (3203)G100010 3360 3353lowast 336731 117148 (3294) 3355 336608 132359 (3267)G100020 7818 7817 782981 201278 (3475) 7815lowast 783048 236369 (3342)U50005 109 112 11639 13829 (1640) 109 11318 12115 (1595)U50010 523 531 53775 22552 (1512) 526 53184 16626 (1222)U50020 1825 1831 184139 28507 (1344) 1823lowast 183188 29947 (1197)U50040 5328 5364 538001 56174 (1381) 5348 536983 52315 (1331)U100005 117 118 12602 45103 (3137) 115lowast 12349 46435 (2967)U100010 577 576 58326 59901 (2599) 571lowast 57816 74350 (2603)U100020 2367 2375 239633 79877 (1963) 2373 238893 93043 (1848)U100040 7329 7399 741749 142178 (1634) 7382 740718 149310 (1573)1Best known values [20ndash22 24 33]2Best and average results from 100 runs Asterisked numbers are new best values3CPU seconds on a Pentium IV 28GHz4Average number of generations from 100 runs

graphs This suggests that effective cluster handling is moreimportant on the geometric graphs as we suggested inSection 31

The local optimization algorithm ismuchmore expensivethan the cluster-handling heuristic thus CMPA does nottake much longer to run than KL-MA or GEFM On averageCMPA required 1414more time than KL-MA for the bipar-titioning problems and 202 more than GEFM in 32-waypartitioning CMPA ran 552 faster in 8-way partitioning

7 Concluding Remarks

We devised a cluster-moving heuristic and combined it witha memetic algorithm (MA) for graph partitioning Experi-ments on 2-way 8-way and 32-way partitioning showed thatthis heuristic significantly improved the performance of theMA especially on the 32-way partitioning

The method of moving clusters that we have introducedaddresses a significant weakness in standard IIP algorithmsThe idea of adding an operation to complement a localoptimization algorithm might be used to address otherdeficiencies in MAs

Our method of cluster detection is based on a measurethat we call genic distance which is designed to reflect theextent to which two vertices connected by an edge belongto the same cluster Instead of computing genic distancesonce in an initialization phase anMA could recompute thesedistances as it progresses this might improve the accuracyof cluster detection at some cost in time We believe thatgenic distance might also be useful in solving other problemsinvolving clusters such as the maximum clique problem

Table 7 Real-world benchmark graphs

Graph Number of vertices Number of edgesnasa4704 4704 50026bcspwr09 1723 2394bcsstk13 2003 40940DEBR12 4096 8189

Appendix

Results on Real-World Graphs

We also tested on four real-world benchmark graphs usedin [11 15 43] The sizes of the graphs are given in Table 7We conducted experiments on 2-way and 8-way partitioningWe performed 100 runs for bipartitioning and 50 runs for8-way partitioning It was run on an Intel Core i7 36GHzcomputer with Linux Table 8 compares CMPA with KL-MAon bipartitioning and Table 9 compares CMPA with GEFMon 8-way partitioning In the tables the bold-faced numbersindicate the lower of the average cut sizes obtained by the twoalgorithms Similarly to the results in Section 6CMPAoverallperformed better than the others with comparable runningtimes

Disclosure

Apreliminary version of this paper appeared inProceedings ofthe Genetic and Evolutionary Computation Conference 2007(p 1520)

Mathematical Problems in Engineering 9

Table 8 Comparison of KL-MA and CMPA on bipartitioning

Graph Best known1 KL-MA CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

nasa4704 1292 1292 129200 415 (407) 1292 129200 382 (347)bcspwr09 9 9 1131 061 (1010) 9 1060 081 (1003)bcsstk13 2355 2355 235500 117 (330) 2355 235500 146 (282)DEBR12 548 548 54844 991 (765) 548 54820 1032 (794)1Best result from the literature2Best and average results from 100 runs3CPU seconds on Intel Core i7 36GHz4Average number of generations over 100 runs

Table 9 Comparison of GEFM and CMPA on 8-way partitioning

Graph GEFM [21] CMPABest1 Ave1 CPU2 (Gen3) Best1 Ave1 CPU2 (Gen3)

nasa4704 3898 390366 57569 (2235) 3896 390266 59151 (2255)bcspwr09 53 5708 7880 (2947) 54 5774 8516 (2925)bcsstk13 8911 893910 28119 (1427) 8919 893614 30154 (1476)DEBR12 1248 126016 31597 (3969) 1248 125990 35383 (4002)1Best and average results from 50 runs2CPU seconds on Intel Core i7 36GHz3Average number of generations over 50 runs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the ICT RampD program ofMSIPIITP Korea (10045253 The development of SDNOpenFlow-Based Enterprise Network Controller Technol-ogy) project The authors would like to thank ProfessorByung-Ro Moon and Dr Yongjoo Song for their valuablesuggestions for improving this paper The authors also thankJong-Pil Kim for providing the source code of GEFM [21]

References

[1] Y Kanizo D Hay and I Keslassy ldquoPalette distributing tablesin software-defined networksrdquo in Proceedings of the IEEEInternational Conference on Computer Communications pp545ndash549 April 2013

[2] P Wette M Draxler and A Schwabe ldquoMaxiNet distributedemulation of software-defined networksrdquo in Proceedings of theIFIP Networking Conference pp 1ndash9 June 2014

[3] M R Garey and D S Johnson Computers and Intractability AGuide to the Theory of NP-Completeness W H Freeman 1979

[4] T N Bui and C Jones ldquoFinding good approximate vertex andedge partitions is NP-hardrdquo Information Processing Letters vol42 no 3 pp 153ndash159 1992

[5] Y-H Kim ldquoImproved implementation choices for iterativeimprovement partitioning algorithms on circuitsrdquo in Proceed-ings of the International Conference on Computer Design pp 30ndash34 July 2008

[6] Y Yoon and Y-H Kim ldquoNew bucket managements in iterativeimprovement partitioning algorithmsrdquo Applied Mathematics ampInformation Sciences vol 7 no 2 pp 529ndash532 2013

[7] BW Kernighan and S Lin ldquoAn efficient heuristic procedure forpartitioning graphsrdquo Bell System Technical Journal vol 49 no2 pp 291ndash307 1970

[8] C Fiduccia and R Mattheyses ldquoA Linear-time heuristic forimproving network partitionsrdquo in Proceedings of the Conferenceon Design Automation pp 175ndash181 June 1982

[9] D S Johnson C R Aragon L A McGeoch and C SchevonldquoOptimization by simulated annealing An experimental evalu-ation Part I Graph partitioningrdquo Operations Research vol 37no 6 pp 865ndash892 1989

[10] T N Bui and B R Moon ldquoGenetic algorithm and graphpartitioningrdquo IEEETransactions on Computers vol 45 no 7 pp841ndash855 1996

[11] R Battiti and A A Bertossi ldquoGreedy prohibition and reactiveheuristics for graph partitioningrdquo IEEE Transactions on Com-puters vol 48 no 4 pp 361ndash385 1999

[12] E Rolland H Pirkul and F Glover ldquoTabu search for graphpartitioningrdquo Annals of Operations Research vol 63 no 2 pp209ndash232 1996

[13] T N Bui and L C Strite ldquoAn ant system algorithm for graphbisectionrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference pp 43ndash51 July 2002

[14] I Hwang Y-H Kim and B-R Moon ldquoMulti-attractor genereordering for graph bisectionrdquo inProceedings of theGenetic andEvolutionary Computation Conference pp 1209ndash1215 July 2006

[15] Y-H Kim and B-R Moon ldquoLock gain based graph partition-ingrdquo Journal of Heuristics vol 10 no 1 pp 37ndash57 2004

[16] P Merz and B Freisleben ldquoFitness landscapes memetic algo-rithms and greedy operators for graph bipartitioningrdquo Evolu-tionary Computation vol 8 no 1 pp 61ndash91 2000

[17] HMuhlenbein and TMahnig ldquoEvolutionary optimization andthe estimation of search distributions with applications to graph

10 Mathematical Problems in Engineering

bipartitioningrdquo International Journal of Approximate Reasoningvol 31 no 3 pp 157ndash192 2002

[18] A G Steenbeek E Marchiori and A E Eiben ldquoFindingbalanced graph bi-partitions using a hybrid genetic algorithmrdquoin Proceedings of the IEEE International Conference on Evolu-tionary Computation pp 90ndash95 May 1998

[19] Y Yoon and Y-H Kim ldquoVertex ordering clustering andtheir application to graph partitioningrdquoApplied Mathematics ampInformation Sciences vol 8 no 1 pp 135ndash138 2014

[20] S-J Kang and B-R Moon ldquoA hybrid genetic algorithm formultiway graph partitioningrdquo in Proceedings of the Genetic andEvolutionary Computation Conference pp 159ndash166 July 2000

[21] J-P Kim and B-R Moon ldquoA hybrid genetic search for multi-way graph partitioning based on direct partitioningrdquo in Pro-ceedings of the Genetic and Evolutionary Computation Confer-ence pp 408ndash415 July 2001

[22] Y-H Kim A Moraglio Y Yoon and B-R Moon ldquoGeometriccrossover for multiway graph partitioningrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1217ndash1224 July 2006

[23] A Moraglio Y-H Kim Y Yoon and B-R Moon ldquoGeometriccrossovers for multiway graph partitioningrdquo Evolutionary Com-putation vol 15 no 4 pp 445ndash474 2007

[24] AMoraglio Y-H Kim Y Yoon B-RMoon and R Poli ldquoGeo-metric crossover for permutations with repetitions applicationto graph partitioningrdquo in Proceedings of the PPSN Workshopon Evolutionary AlgorithmsmdashBridging Theory and PracticeSeptember 2006

[25] A J Soper C Walshaw and M Cross ldquoA combined evolu-tionary search and multilevel optimisation approach to graph-partitioningrdquo Journal of Global Optimization vol 29 no 2 pp225ndash241 2004

[26] Y Yoon Y-H Kim A Moraglio and B-R Moon ldquoQuotientgeometric crossovers and redundant encodingsrdquo TheoreticalComputer Science vol 425 pp 4ndash16 2012

[27] J Kim I Hwang Y-H Kim and B-R Moon ldquoGeneticapproaches for graph partitioning a surveyrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 473ndash480 July 2011

[28] S Dutt and W Deng ldquoA probability-based approach to VLSIcircuit partitioningrdquo in Proceedings of the Design AutomationConference pp 100ndash105 June 1996

[29] B Krishnamurthy ldquoAn improved min-cut algorithm for parti-tioning VLSI networksrdquo IEEE Transactions on Computers vol33 no 5 pp 438ndash446 1984

[30] L A Sanchis ldquoMultiple-way network partitioningrdquo IEEETrans-actions on Computers vol 38 no 1 pp 62ndash81 1989

[31] G Laszewski ldquoIntelligent structural operators for the k-waygraph partitioning problemrdquo in Proceedings of the InternationalConference on Genetic Algorithms pp 45ndash52 July 1991

[32] H Muhlenbein ldquoParallel genetic algorithms in combinatorialoptimizationrdquo in Computer Science and Operations Research OBalci R Sharda and S Zenios Eds pp 441ndash453 PergamonOxford UK 1992

[33] S-S Choi and B-R Moon ldquoNormalization in genetic algo-rithmsrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference vol 2723 of Lecture Notes in ComputerScience pp 862ndash873 2003

[34] S-H Kim Y-H Kim and B-R Moon ldquoA hybrid geneticalgorithm for themax cut problemrdquo inProceedings of theGeneticand Evolutionary Computation Conference pp 416ndash423 July2001

[35] Y-H Kim Y-K Kwon and B-R Moon ldquoProblem-independ-ent schema synthesis for genetic algorithmsrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1112ndash1122 July 2003

[36] K Seo S Hyun and Y-H Kim ldquoAn edge-set representationbased on spanning tree for searching cut spacerdquo IEEE Trans-actions on Evolutionary Computation 2014

[37] Y-H Kim Combinatorial optimization based on effective localsearch genetic algorithms and problem space analyses [PhDthesis] Seoul National University Seoul Republic of Korea2005

[38] Y-H Kim ldquoAn enzyme-inspired approach to surmount barriersin graph bisectionrdquo in Proceedings of the International Confer-ence on Computational Science and Its Applications vol 5072 ofLecture Notes in Computer Science pp 841ndash851 2008

[39] S Dutt and W Deng ldquoVLSI circuit partitioning by cluster-removal using iterative improvement techniquesrdquo in Proceed-ings of the IEEEACM International Conference on Computer-Aided Design pp 194ndash200 November 1996

[40] S Dutt and W Deng ldquoCluster-aware iterative improvementtechniques for partitioning large VLSI circuitsrdquo ACM Transac-tions on Design Automation of Electronic Systems vol 7 no 1pp 91ndash121 2002

[41] Y Yoon and Y-H Kim ldquoAn efficient genetic algorithm formaximum coverage deployment in wireless sensor networksrdquoIEEE Transactions on Cybernetics vol 43 no 5 pp 1473ndash14832013

[42] Y Yoon Y-H Kim AMoraglio and B-RMoon ldquoA theoreticaland empirical study on unbiased boundary-extended crossoverfor real-valued representationrdquo Information Sciences vol 183no 1 pp 48ndash65 2012

[43] BMonien andRDiekmann ldquoA local graph partitioning heuris-tic meeting bisection boundsrdquo in Proceedings of the SIAM Con-ference on Parallel Processing for Scientific Computing March1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Table 4 Comparison of KL-MA and CMPA on bipartitioning

Graph Best known1 KL-MA CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

G50025 49 49 5134 032 (595) 49 5122 034 (558)G50005 218 218 21865 059 (700) 218 21845 064 (682)G50010 626 626 62755 102 (681) 626 62746 110 (672)G50020 1744 1744 174583 193 (694) 1744 174562 209 (687)G100025 93 93 9726 096 (745) 93 9708 104 (722)G100005 445 445 45150 207 (884) 445 45092 222 (860)G100010 1362 1362 136668 407 (969) 1362 136618 445 (977)G100020 3382 3382 338561 742 (971) 3382 338526 813 (987)U50005 2 2 496 055 (793) 2 394 066 (803)U50010 26 26 2631 045 (459) 26 2604 057 (465)U50020 178 178 17806 042 (300) 178 17800 053 (271)U50040 412 412 41200 034 (190) 412 41200 037 (102)U100005 1 1 1167 171 (1077) 1 881 212 (1068)U100010 39 39 4627 151 (645) 39 4415 195 (653)U100020 222 222 22219 113 (347) 222 22207 151 (327)U100040 737 737 73700 120 (243) 737 73700 177 (205)1Best result from the literature2Best and average results from 1000 runs3CPU seconds on a Pentium IV 28GHz4Average number of generations over 1000 runs

Table 5 Comparison of GEFM and CMPA on 8-way partitioning

Graph GEFM [21] CMPABest1 Ave1 CPU2 (Gen3) Best1 Ave1 CPU2 (Gen3)

G50025 111 11521 4203 (2090) 111 11451 4025 (2091)G50005 465 46816 8565 (2314) 465 46763 7296 (2239)G50010 1254 125953 14454 (2347) 1254 125847 13504 (2297)G50020 3348 335480 29318 (2437) 3348 335371 28509 (2317)G100025 212 21630 14676 (3043) 211 21657 14842 (3023)G100005 930 93811 24720 (3148) 931 93926 23467 (3069)G100010 2714 272633 40896 (3146) 2711 272552 41698 (3083)G100020 6525 653655 79129 (3303) 6520 653835 78167 (3141)U50005 16 1880 7087 (2629) 16 1722 5589 (2137)U50010 143 14512 9875 (2277) 143 14426 9050 (1984)U50020 612 61457 12122 (1690) 611 61293 9753 (1483)U50040 1867 187260 15794 (1396) 1867 187173 12962 (1204)U100005 21 3366 18035 (3638) 20 2853 20953 (3172)U100010 176 18370 24308 (3105) 176 18230 25702 (2857)U100020 812 81405 23108 (1824) 812 81300 23293 (1756)U100040 2562 256305 24946 (1353) 2562 256285 27351 (1238)1Best and average results from 100 runs2CPU seconds on a Pentium IV 28GHz3Average number of generations over 100 runs

heuristic but KL-MA andGEFM do notWe performed 1000runs for bipartitioning and 100 runs for 8-way and 32-waypartitioning All the programs were written in the C languageand compiled using GNUrsquos gcc compiler It was run on aPentium IV 28GHz computer with Linux In the tables thebold-faced numbers indicate the lower of the average cut sizes

obtained by the two algorithms CMPA outperformed KL-MA and GEFM on most graphs with comparable runningtimes

CMPA underperformed on some random graphs whichmay be due to the weak cluster structures in these graphsCMPArsquos average performance was better on all the geometric

8 Mathematical Problems in Engineering

Table 6 Comparison of GEFM and CMPA on 32-way partitioning

Graph Best known1 GEFM [21] CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

G50025 177 177 18169 7898 (1491) 177 18082 7430 (1382)G50005 623 624 63047 16226 (2197) 623 62992 15918 (2035)G50010 1573 1574 158194 30083 (2364) 1572lowast 157972 35573 (2332)G50020 4034 4037 404506 68153 (2565) 4035 404281 67319 (2479)G100025 312 313 32099 33579 (2798) 313 32025 35644 (2648)G100005 1217 1208 121872 69867 (3133) 1205lowast 121689 73806 (3203)G100010 3360 3353lowast 336731 117148 (3294) 3355 336608 132359 (3267)G100020 7818 7817 782981 201278 (3475) 7815lowast 783048 236369 (3342)U50005 109 112 11639 13829 (1640) 109 11318 12115 (1595)U50010 523 531 53775 22552 (1512) 526 53184 16626 (1222)U50020 1825 1831 184139 28507 (1344) 1823lowast 183188 29947 (1197)U50040 5328 5364 538001 56174 (1381) 5348 536983 52315 (1331)U100005 117 118 12602 45103 (3137) 115lowast 12349 46435 (2967)U100010 577 576 58326 59901 (2599) 571lowast 57816 74350 (2603)U100020 2367 2375 239633 79877 (1963) 2373 238893 93043 (1848)U100040 7329 7399 741749 142178 (1634) 7382 740718 149310 (1573)1Best known values [20ndash22 24 33]2Best and average results from 100 runs Asterisked numbers are new best values3CPU seconds on a Pentium IV 28GHz4Average number of generations from 100 runs

graphs This suggests that effective cluster handling is moreimportant on the geometric graphs as we suggested inSection 31

The local optimization algorithm ismuchmore expensivethan the cluster-handling heuristic thus CMPA does nottake much longer to run than KL-MA or GEFM On averageCMPA required 1414more time than KL-MA for the bipar-titioning problems and 202 more than GEFM in 32-waypartitioning CMPA ran 552 faster in 8-way partitioning

7 Concluding Remarks

We devised a cluster-moving heuristic and combined it witha memetic algorithm (MA) for graph partitioning Experi-ments on 2-way 8-way and 32-way partitioning showed thatthis heuristic significantly improved the performance of theMA especially on the 32-way partitioning

The method of moving clusters that we have introducedaddresses a significant weakness in standard IIP algorithmsThe idea of adding an operation to complement a localoptimization algorithm might be used to address otherdeficiencies in MAs

Our method of cluster detection is based on a measurethat we call genic distance which is designed to reflect theextent to which two vertices connected by an edge belongto the same cluster Instead of computing genic distancesonce in an initialization phase anMA could recompute thesedistances as it progresses this might improve the accuracyof cluster detection at some cost in time We believe thatgenic distance might also be useful in solving other problemsinvolving clusters such as the maximum clique problem

Table 7 Real-world benchmark graphs

Graph Number of vertices Number of edgesnasa4704 4704 50026bcspwr09 1723 2394bcsstk13 2003 40940DEBR12 4096 8189

Appendix

Results on Real-World Graphs

We also tested on four real-world benchmark graphs usedin [11 15 43] The sizes of the graphs are given in Table 7We conducted experiments on 2-way and 8-way partitioningWe performed 100 runs for bipartitioning and 50 runs for8-way partitioning It was run on an Intel Core i7 36GHzcomputer with Linux Table 8 compares CMPA with KL-MAon bipartitioning and Table 9 compares CMPA with GEFMon 8-way partitioning In the tables the bold-faced numbersindicate the lower of the average cut sizes obtained by the twoalgorithms Similarly to the results in Section 6CMPAoverallperformed better than the others with comparable runningtimes

Disclosure

Apreliminary version of this paper appeared inProceedings ofthe Genetic and Evolutionary Computation Conference 2007(p 1520)

Mathematical Problems in Engineering 9

Table 8 Comparison of KL-MA and CMPA on bipartitioning

Graph Best known1 KL-MA CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

nasa4704 1292 1292 129200 415 (407) 1292 129200 382 (347)bcspwr09 9 9 1131 061 (1010) 9 1060 081 (1003)bcsstk13 2355 2355 235500 117 (330) 2355 235500 146 (282)DEBR12 548 548 54844 991 (765) 548 54820 1032 (794)1Best result from the literature2Best and average results from 100 runs3CPU seconds on Intel Core i7 36GHz4Average number of generations over 100 runs

Table 9 Comparison of GEFM and CMPA on 8-way partitioning

Graph GEFM [21] CMPABest1 Ave1 CPU2 (Gen3) Best1 Ave1 CPU2 (Gen3)

nasa4704 3898 390366 57569 (2235) 3896 390266 59151 (2255)bcspwr09 53 5708 7880 (2947) 54 5774 8516 (2925)bcsstk13 8911 893910 28119 (1427) 8919 893614 30154 (1476)DEBR12 1248 126016 31597 (3969) 1248 125990 35383 (4002)1Best and average results from 50 runs2CPU seconds on Intel Core i7 36GHz3Average number of generations over 50 runs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the ICT RampD program ofMSIPIITP Korea (10045253 The development of SDNOpenFlow-Based Enterprise Network Controller Technol-ogy) project The authors would like to thank ProfessorByung-Ro Moon and Dr Yongjoo Song for their valuablesuggestions for improving this paper The authors also thankJong-Pil Kim for providing the source code of GEFM [21]

References

[1] Y Kanizo D Hay and I Keslassy ldquoPalette distributing tablesin software-defined networksrdquo in Proceedings of the IEEEInternational Conference on Computer Communications pp545ndash549 April 2013

[2] P Wette M Draxler and A Schwabe ldquoMaxiNet distributedemulation of software-defined networksrdquo in Proceedings of theIFIP Networking Conference pp 1ndash9 June 2014

[3] M R Garey and D S Johnson Computers and Intractability AGuide to the Theory of NP-Completeness W H Freeman 1979

[4] T N Bui and C Jones ldquoFinding good approximate vertex andedge partitions is NP-hardrdquo Information Processing Letters vol42 no 3 pp 153ndash159 1992

[5] Y-H Kim ldquoImproved implementation choices for iterativeimprovement partitioning algorithms on circuitsrdquo in Proceed-ings of the International Conference on Computer Design pp 30ndash34 July 2008

[6] Y Yoon and Y-H Kim ldquoNew bucket managements in iterativeimprovement partitioning algorithmsrdquo Applied Mathematics ampInformation Sciences vol 7 no 2 pp 529ndash532 2013

[7] BW Kernighan and S Lin ldquoAn efficient heuristic procedure forpartitioning graphsrdquo Bell System Technical Journal vol 49 no2 pp 291ndash307 1970

[8] C Fiduccia and R Mattheyses ldquoA Linear-time heuristic forimproving network partitionsrdquo in Proceedings of the Conferenceon Design Automation pp 175ndash181 June 1982

[9] D S Johnson C R Aragon L A McGeoch and C SchevonldquoOptimization by simulated annealing An experimental evalu-ation Part I Graph partitioningrdquo Operations Research vol 37no 6 pp 865ndash892 1989

[10] T N Bui and B R Moon ldquoGenetic algorithm and graphpartitioningrdquo IEEETransactions on Computers vol 45 no 7 pp841ndash855 1996

[11] R Battiti and A A Bertossi ldquoGreedy prohibition and reactiveheuristics for graph partitioningrdquo IEEE Transactions on Com-puters vol 48 no 4 pp 361ndash385 1999

[12] E Rolland H Pirkul and F Glover ldquoTabu search for graphpartitioningrdquo Annals of Operations Research vol 63 no 2 pp209ndash232 1996

[13] T N Bui and L C Strite ldquoAn ant system algorithm for graphbisectionrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference pp 43ndash51 July 2002

[14] I Hwang Y-H Kim and B-R Moon ldquoMulti-attractor genereordering for graph bisectionrdquo inProceedings of theGenetic andEvolutionary Computation Conference pp 1209ndash1215 July 2006

[15] Y-H Kim and B-R Moon ldquoLock gain based graph partition-ingrdquo Journal of Heuristics vol 10 no 1 pp 37ndash57 2004

[16] P Merz and B Freisleben ldquoFitness landscapes memetic algo-rithms and greedy operators for graph bipartitioningrdquo Evolu-tionary Computation vol 8 no 1 pp 61ndash91 2000

[17] HMuhlenbein and TMahnig ldquoEvolutionary optimization andthe estimation of search distributions with applications to graph

10 Mathematical Problems in Engineering

bipartitioningrdquo International Journal of Approximate Reasoningvol 31 no 3 pp 157ndash192 2002

[18] A G Steenbeek E Marchiori and A E Eiben ldquoFindingbalanced graph bi-partitions using a hybrid genetic algorithmrdquoin Proceedings of the IEEE International Conference on Evolu-tionary Computation pp 90ndash95 May 1998

[19] Y Yoon and Y-H Kim ldquoVertex ordering clustering andtheir application to graph partitioningrdquoApplied Mathematics ampInformation Sciences vol 8 no 1 pp 135ndash138 2014

[20] S-J Kang and B-R Moon ldquoA hybrid genetic algorithm formultiway graph partitioningrdquo in Proceedings of the Genetic andEvolutionary Computation Conference pp 159ndash166 July 2000

[21] J-P Kim and B-R Moon ldquoA hybrid genetic search for multi-way graph partitioning based on direct partitioningrdquo in Pro-ceedings of the Genetic and Evolutionary Computation Confer-ence pp 408ndash415 July 2001

[22] Y-H Kim A Moraglio Y Yoon and B-R Moon ldquoGeometriccrossover for multiway graph partitioningrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1217ndash1224 July 2006

[23] A Moraglio Y-H Kim Y Yoon and B-R Moon ldquoGeometriccrossovers for multiway graph partitioningrdquo Evolutionary Com-putation vol 15 no 4 pp 445ndash474 2007

[24] AMoraglio Y-H Kim Y Yoon B-RMoon and R Poli ldquoGeo-metric crossover for permutations with repetitions applicationto graph partitioningrdquo in Proceedings of the PPSN Workshopon Evolutionary AlgorithmsmdashBridging Theory and PracticeSeptember 2006

[25] A J Soper C Walshaw and M Cross ldquoA combined evolu-tionary search and multilevel optimisation approach to graph-partitioningrdquo Journal of Global Optimization vol 29 no 2 pp225ndash241 2004

[26] Y Yoon Y-H Kim A Moraglio and B-R Moon ldquoQuotientgeometric crossovers and redundant encodingsrdquo TheoreticalComputer Science vol 425 pp 4ndash16 2012

[27] J Kim I Hwang Y-H Kim and B-R Moon ldquoGeneticapproaches for graph partitioning a surveyrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 473ndash480 July 2011

[28] S Dutt and W Deng ldquoA probability-based approach to VLSIcircuit partitioningrdquo in Proceedings of the Design AutomationConference pp 100ndash105 June 1996

[29] B Krishnamurthy ldquoAn improved min-cut algorithm for parti-tioning VLSI networksrdquo IEEE Transactions on Computers vol33 no 5 pp 438ndash446 1984

[30] L A Sanchis ldquoMultiple-way network partitioningrdquo IEEETrans-actions on Computers vol 38 no 1 pp 62ndash81 1989

[31] G Laszewski ldquoIntelligent structural operators for the k-waygraph partitioning problemrdquo in Proceedings of the InternationalConference on Genetic Algorithms pp 45ndash52 July 1991

[32] H Muhlenbein ldquoParallel genetic algorithms in combinatorialoptimizationrdquo in Computer Science and Operations Research OBalci R Sharda and S Zenios Eds pp 441ndash453 PergamonOxford UK 1992

[33] S-S Choi and B-R Moon ldquoNormalization in genetic algo-rithmsrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference vol 2723 of Lecture Notes in ComputerScience pp 862ndash873 2003

[34] S-H Kim Y-H Kim and B-R Moon ldquoA hybrid geneticalgorithm for themax cut problemrdquo inProceedings of theGeneticand Evolutionary Computation Conference pp 416ndash423 July2001

[35] Y-H Kim Y-K Kwon and B-R Moon ldquoProblem-independ-ent schema synthesis for genetic algorithmsrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1112ndash1122 July 2003

[36] K Seo S Hyun and Y-H Kim ldquoAn edge-set representationbased on spanning tree for searching cut spacerdquo IEEE Trans-actions on Evolutionary Computation 2014

[37] Y-H Kim Combinatorial optimization based on effective localsearch genetic algorithms and problem space analyses [PhDthesis] Seoul National University Seoul Republic of Korea2005

[38] Y-H Kim ldquoAn enzyme-inspired approach to surmount barriersin graph bisectionrdquo in Proceedings of the International Confer-ence on Computational Science and Its Applications vol 5072 ofLecture Notes in Computer Science pp 841ndash851 2008

[39] S Dutt and W Deng ldquoVLSI circuit partitioning by cluster-removal using iterative improvement techniquesrdquo in Proceed-ings of the IEEEACM International Conference on Computer-Aided Design pp 194ndash200 November 1996

[40] S Dutt and W Deng ldquoCluster-aware iterative improvementtechniques for partitioning large VLSI circuitsrdquo ACM Transac-tions on Design Automation of Electronic Systems vol 7 no 1pp 91ndash121 2002

[41] Y Yoon and Y-H Kim ldquoAn efficient genetic algorithm formaximum coverage deployment in wireless sensor networksrdquoIEEE Transactions on Cybernetics vol 43 no 5 pp 1473ndash14832013

[42] Y Yoon Y-H Kim AMoraglio and B-RMoon ldquoA theoreticaland empirical study on unbiased boundary-extended crossoverfor real-valued representationrdquo Information Sciences vol 183no 1 pp 48ndash65 2012

[43] BMonien andRDiekmann ldquoA local graph partitioning heuris-tic meeting bisection boundsrdquo in Proceedings of the SIAM Con-ference on Parallel Processing for Scientific Computing March1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 6 Comparison of GEFM and CMPA on 32-way partitioning

Graph Best known1 GEFM [21] CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

G50025 177 177 18169 7898 (1491) 177 18082 7430 (1382)G50005 623 624 63047 16226 (2197) 623 62992 15918 (2035)G50010 1573 1574 158194 30083 (2364) 1572lowast 157972 35573 (2332)G50020 4034 4037 404506 68153 (2565) 4035 404281 67319 (2479)G100025 312 313 32099 33579 (2798) 313 32025 35644 (2648)G100005 1217 1208 121872 69867 (3133) 1205lowast 121689 73806 (3203)G100010 3360 3353lowast 336731 117148 (3294) 3355 336608 132359 (3267)G100020 7818 7817 782981 201278 (3475) 7815lowast 783048 236369 (3342)U50005 109 112 11639 13829 (1640) 109 11318 12115 (1595)U50010 523 531 53775 22552 (1512) 526 53184 16626 (1222)U50020 1825 1831 184139 28507 (1344) 1823lowast 183188 29947 (1197)U50040 5328 5364 538001 56174 (1381) 5348 536983 52315 (1331)U100005 117 118 12602 45103 (3137) 115lowast 12349 46435 (2967)U100010 577 576 58326 59901 (2599) 571lowast 57816 74350 (2603)U100020 2367 2375 239633 79877 (1963) 2373 238893 93043 (1848)U100040 7329 7399 741749 142178 (1634) 7382 740718 149310 (1573)1Best known values [20ndash22 24 33]2Best and average results from 100 runs Asterisked numbers are new best values3CPU seconds on a Pentium IV 28GHz4Average number of generations from 100 runs

graphs This suggests that effective cluster handling is moreimportant on the geometric graphs as we suggested inSection 31

The local optimization algorithm ismuchmore expensivethan the cluster-handling heuristic thus CMPA does nottake much longer to run than KL-MA or GEFM On averageCMPA required 1414more time than KL-MA for the bipar-titioning problems and 202 more than GEFM in 32-waypartitioning CMPA ran 552 faster in 8-way partitioning

7 Concluding Remarks

We devised a cluster-moving heuristic and combined it witha memetic algorithm (MA) for graph partitioning Experi-ments on 2-way 8-way and 32-way partitioning showed thatthis heuristic significantly improved the performance of theMA especially on the 32-way partitioning

The method of moving clusters that we have introducedaddresses a significant weakness in standard IIP algorithmsThe idea of adding an operation to complement a localoptimization algorithm might be used to address otherdeficiencies in MAs

Our method of cluster detection is based on a measurethat we call genic distance which is designed to reflect theextent to which two vertices connected by an edge belongto the same cluster Instead of computing genic distancesonce in an initialization phase anMA could recompute thesedistances as it progresses this might improve the accuracyof cluster detection at some cost in time We believe thatgenic distance might also be useful in solving other problemsinvolving clusters such as the maximum clique problem

Table 7 Real-world benchmark graphs

Graph Number of vertices Number of edgesnasa4704 4704 50026bcspwr09 1723 2394bcsstk13 2003 40940DEBR12 4096 8189

Appendix

Results on Real-World Graphs

We also tested on four real-world benchmark graphs usedin [11 15 43] The sizes of the graphs are given in Table 7We conducted experiments on 2-way and 8-way partitioningWe performed 100 runs for bipartitioning and 50 runs for8-way partitioning It was run on an Intel Core i7 36GHzcomputer with Linux Table 8 compares CMPA with KL-MAon bipartitioning and Table 9 compares CMPA with GEFMon 8-way partitioning In the tables the bold-faced numbersindicate the lower of the average cut sizes obtained by the twoalgorithms Similarly to the results in Section 6CMPAoverallperformed better than the others with comparable runningtimes

Disclosure

Apreliminary version of this paper appeared inProceedings ofthe Genetic and Evolutionary Computation Conference 2007(p 1520)

Mathematical Problems in Engineering 9

Table 8 Comparison of KL-MA and CMPA on bipartitioning

Graph Best known1 KL-MA CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

nasa4704 1292 1292 129200 415 (407) 1292 129200 382 (347)bcspwr09 9 9 1131 061 (1010) 9 1060 081 (1003)bcsstk13 2355 2355 235500 117 (330) 2355 235500 146 (282)DEBR12 548 548 54844 991 (765) 548 54820 1032 (794)1Best result from the literature2Best and average results from 100 runs3CPU seconds on Intel Core i7 36GHz4Average number of generations over 100 runs

Table 9 Comparison of GEFM and CMPA on 8-way partitioning

Graph GEFM [21] CMPABest1 Ave1 CPU2 (Gen3) Best1 Ave1 CPU2 (Gen3)

nasa4704 3898 390366 57569 (2235) 3896 390266 59151 (2255)bcspwr09 53 5708 7880 (2947) 54 5774 8516 (2925)bcsstk13 8911 893910 28119 (1427) 8919 893614 30154 (1476)DEBR12 1248 126016 31597 (3969) 1248 125990 35383 (4002)1Best and average results from 50 runs2CPU seconds on Intel Core i7 36GHz3Average number of generations over 50 runs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the ICT RampD program ofMSIPIITP Korea (10045253 The development of SDNOpenFlow-Based Enterprise Network Controller Technol-ogy) project The authors would like to thank ProfessorByung-Ro Moon and Dr Yongjoo Song for their valuablesuggestions for improving this paper The authors also thankJong-Pil Kim for providing the source code of GEFM [21]

References

[1] Y Kanizo D Hay and I Keslassy ldquoPalette distributing tablesin software-defined networksrdquo in Proceedings of the IEEEInternational Conference on Computer Communications pp545ndash549 April 2013

[2] P Wette M Draxler and A Schwabe ldquoMaxiNet distributedemulation of software-defined networksrdquo in Proceedings of theIFIP Networking Conference pp 1ndash9 June 2014

[3] M R Garey and D S Johnson Computers and Intractability AGuide to the Theory of NP-Completeness W H Freeman 1979

[4] T N Bui and C Jones ldquoFinding good approximate vertex andedge partitions is NP-hardrdquo Information Processing Letters vol42 no 3 pp 153ndash159 1992

[5] Y-H Kim ldquoImproved implementation choices for iterativeimprovement partitioning algorithms on circuitsrdquo in Proceed-ings of the International Conference on Computer Design pp 30ndash34 July 2008

[6] Y Yoon and Y-H Kim ldquoNew bucket managements in iterativeimprovement partitioning algorithmsrdquo Applied Mathematics ampInformation Sciences vol 7 no 2 pp 529ndash532 2013

[7] BW Kernighan and S Lin ldquoAn efficient heuristic procedure forpartitioning graphsrdquo Bell System Technical Journal vol 49 no2 pp 291ndash307 1970

[8] C Fiduccia and R Mattheyses ldquoA Linear-time heuristic forimproving network partitionsrdquo in Proceedings of the Conferenceon Design Automation pp 175ndash181 June 1982

[9] D S Johnson C R Aragon L A McGeoch and C SchevonldquoOptimization by simulated annealing An experimental evalu-ation Part I Graph partitioningrdquo Operations Research vol 37no 6 pp 865ndash892 1989

[10] T N Bui and B R Moon ldquoGenetic algorithm and graphpartitioningrdquo IEEETransactions on Computers vol 45 no 7 pp841ndash855 1996

[11] R Battiti and A A Bertossi ldquoGreedy prohibition and reactiveheuristics for graph partitioningrdquo IEEE Transactions on Com-puters vol 48 no 4 pp 361ndash385 1999

[12] E Rolland H Pirkul and F Glover ldquoTabu search for graphpartitioningrdquo Annals of Operations Research vol 63 no 2 pp209ndash232 1996

[13] T N Bui and L C Strite ldquoAn ant system algorithm for graphbisectionrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference pp 43ndash51 July 2002

[14] I Hwang Y-H Kim and B-R Moon ldquoMulti-attractor genereordering for graph bisectionrdquo inProceedings of theGenetic andEvolutionary Computation Conference pp 1209ndash1215 July 2006

[15] Y-H Kim and B-R Moon ldquoLock gain based graph partition-ingrdquo Journal of Heuristics vol 10 no 1 pp 37ndash57 2004

[16] P Merz and B Freisleben ldquoFitness landscapes memetic algo-rithms and greedy operators for graph bipartitioningrdquo Evolu-tionary Computation vol 8 no 1 pp 61ndash91 2000

[17] HMuhlenbein and TMahnig ldquoEvolutionary optimization andthe estimation of search distributions with applications to graph

10 Mathematical Problems in Engineering

bipartitioningrdquo International Journal of Approximate Reasoningvol 31 no 3 pp 157ndash192 2002

[18] A G Steenbeek E Marchiori and A E Eiben ldquoFindingbalanced graph bi-partitions using a hybrid genetic algorithmrdquoin Proceedings of the IEEE International Conference on Evolu-tionary Computation pp 90ndash95 May 1998

[19] Y Yoon and Y-H Kim ldquoVertex ordering clustering andtheir application to graph partitioningrdquoApplied Mathematics ampInformation Sciences vol 8 no 1 pp 135ndash138 2014

[20] S-J Kang and B-R Moon ldquoA hybrid genetic algorithm formultiway graph partitioningrdquo in Proceedings of the Genetic andEvolutionary Computation Conference pp 159ndash166 July 2000

[21] J-P Kim and B-R Moon ldquoA hybrid genetic search for multi-way graph partitioning based on direct partitioningrdquo in Pro-ceedings of the Genetic and Evolutionary Computation Confer-ence pp 408ndash415 July 2001

[22] Y-H Kim A Moraglio Y Yoon and B-R Moon ldquoGeometriccrossover for multiway graph partitioningrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1217ndash1224 July 2006

[23] A Moraglio Y-H Kim Y Yoon and B-R Moon ldquoGeometriccrossovers for multiway graph partitioningrdquo Evolutionary Com-putation vol 15 no 4 pp 445ndash474 2007

[24] AMoraglio Y-H Kim Y Yoon B-RMoon and R Poli ldquoGeo-metric crossover for permutations with repetitions applicationto graph partitioningrdquo in Proceedings of the PPSN Workshopon Evolutionary AlgorithmsmdashBridging Theory and PracticeSeptember 2006

[25] A J Soper C Walshaw and M Cross ldquoA combined evolu-tionary search and multilevel optimisation approach to graph-partitioningrdquo Journal of Global Optimization vol 29 no 2 pp225ndash241 2004

[26] Y Yoon Y-H Kim A Moraglio and B-R Moon ldquoQuotientgeometric crossovers and redundant encodingsrdquo TheoreticalComputer Science vol 425 pp 4ndash16 2012

[27] J Kim I Hwang Y-H Kim and B-R Moon ldquoGeneticapproaches for graph partitioning a surveyrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 473ndash480 July 2011

[28] S Dutt and W Deng ldquoA probability-based approach to VLSIcircuit partitioningrdquo in Proceedings of the Design AutomationConference pp 100ndash105 June 1996

[29] B Krishnamurthy ldquoAn improved min-cut algorithm for parti-tioning VLSI networksrdquo IEEE Transactions on Computers vol33 no 5 pp 438ndash446 1984

[30] L A Sanchis ldquoMultiple-way network partitioningrdquo IEEETrans-actions on Computers vol 38 no 1 pp 62ndash81 1989

[31] G Laszewski ldquoIntelligent structural operators for the k-waygraph partitioning problemrdquo in Proceedings of the InternationalConference on Genetic Algorithms pp 45ndash52 July 1991

[32] H Muhlenbein ldquoParallel genetic algorithms in combinatorialoptimizationrdquo in Computer Science and Operations Research OBalci R Sharda and S Zenios Eds pp 441ndash453 PergamonOxford UK 1992

[33] S-S Choi and B-R Moon ldquoNormalization in genetic algo-rithmsrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference vol 2723 of Lecture Notes in ComputerScience pp 862ndash873 2003

[34] S-H Kim Y-H Kim and B-R Moon ldquoA hybrid geneticalgorithm for themax cut problemrdquo inProceedings of theGeneticand Evolutionary Computation Conference pp 416ndash423 July2001

[35] Y-H Kim Y-K Kwon and B-R Moon ldquoProblem-independ-ent schema synthesis for genetic algorithmsrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1112ndash1122 July 2003

[36] K Seo S Hyun and Y-H Kim ldquoAn edge-set representationbased on spanning tree for searching cut spacerdquo IEEE Trans-actions on Evolutionary Computation 2014

[37] Y-H Kim Combinatorial optimization based on effective localsearch genetic algorithms and problem space analyses [PhDthesis] Seoul National University Seoul Republic of Korea2005

[38] Y-H Kim ldquoAn enzyme-inspired approach to surmount barriersin graph bisectionrdquo in Proceedings of the International Confer-ence on Computational Science and Its Applications vol 5072 ofLecture Notes in Computer Science pp 841ndash851 2008

[39] S Dutt and W Deng ldquoVLSI circuit partitioning by cluster-removal using iterative improvement techniquesrdquo in Proceed-ings of the IEEEACM International Conference on Computer-Aided Design pp 194ndash200 November 1996

[40] S Dutt and W Deng ldquoCluster-aware iterative improvementtechniques for partitioning large VLSI circuitsrdquo ACM Transac-tions on Design Automation of Electronic Systems vol 7 no 1pp 91ndash121 2002

[41] Y Yoon and Y-H Kim ldquoAn efficient genetic algorithm formaximum coverage deployment in wireless sensor networksrdquoIEEE Transactions on Cybernetics vol 43 no 5 pp 1473ndash14832013

[42] Y Yoon Y-H Kim AMoraglio and B-RMoon ldquoA theoreticaland empirical study on unbiased boundary-extended crossoverfor real-valued representationrdquo Information Sciences vol 183no 1 pp 48ndash65 2012

[43] BMonien andRDiekmann ldquoA local graph partitioning heuris-tic meeting bisection boundsrdquo in Proceedings of the SIAM Con-ference on Parallel Processing for Scientific Computing March1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

Table 8 Comparison of KL-MA and CMPA on bipartitioning

Graph Best known1 KL-MA CMPABest2 Ave2 CPU3 (Gen4) Best2 Ave2 CPU3 (Gen4)

nasa4704 1292 1292 129200 415 (407) 1292 129200 382 (347)bcspwr09 9 9 1131 061 (1010) 9 1060 081 (1003)bcsstk13 2355 2355 235500 117 (330) 2355 235500 146 (282)DEBR12 548 548 54844 991 (765) 548 54820 1032 (794)1Best result from the literature2Best and average results from 100 runs3CPU seconds on Intel Core i7 36GHz4Average number of generations over 100 runs

Table 9 Comparison of GEFM and CMPA on 8-way partitioning

Graph GEFM [21] CMPABest1 Ave1 CPU2 (Gen3) Best1 Ave1 CPU2 (Gen3)

nasa4704 3898 390366 57569 (2235) 3896 390266 59151 (2255)bcspwr09 53 5708 7880 (2947) 54 5774 8516 (2925)bcsstk13 8911 893910 28119 (1427) 8919 893614 30154 (1476)DEBR12 1248 126016 31597 (3969) 1248 125990 35383 (4002)1Best and average results from 50 runs2CPU seconds on Intel Core i7 36GHz3Average number of generations over 50 runs

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This work was supported by the ICT RampD program ofMSIPIITP Korea (10045253 The development of SDNOpenFlow-Based Enterprise Network Controller Technol-ogy) project The authors would like to thank ProfessorByung-Ro Moon and Dr Yongjoo Song for their valuablesuggestions for improving this paper The authors also thankJong-Pil Kim for providing the source code of GEFM [21]

References

[1] Y Kanizo D Hay and I Keslassy ldquoPalette distributing tablesin software-defined networksrdquo in Proceedings of the IEEEInternational Conference on Computer Communications pp545ndash549 April 2013

[2] P Wette M Draxler and A Schwabe ldquoMaxiNet distributedemulation of software-defined networksrdquo in Proceedings of theIFIP Networking Conference pp 1ndash9 June 2014

[3] M R Garey and D S Johnson Computers and Intractability AGuide to the Theory of NP-Completeness W H Freeman 1979

[4] T N Bui and C Jones ldquoFinding good approximate vertex andedge partitions is NP-hardrdquo Information Processing Letters vol42 no 3 pp 153ndash159 1992

[5] Y-H Kim ldquoImproved implementation choices for iterativeimprovement partitioning algorithms on circuitsrdquo in Proceed-ings of the International Conference on Computer Design pp 30ndash34 July 2008

[6] Y Yoon and Y-H Kim ldquoNew bucket managements in iterativeimprovement partitioning algorithmsrdquo Applied Mathematics ampInformation Sciences vol 7 no 2 pp 529ndash532 2013

[7] BW Kernighan and S Lin ldquoAn efficient heuristic procedure forpartitioning graphsrdquo Bell System Technical Journal vol 49 no2 pp 291ndash307 1970

[8] C Fiduccia and R Mattheyses ldquoA Linear-time heuristic forimproving network partitionsrdquo in Proceedings of the Conferenceon Design Automation pp 175ndash181 June 1982

[9] D S Johnson C R Aragon L A McGeoch and C SchevonldquoOptimization by simulated annealing An experimental evalu-ation Part I Graph partitioningrdquo Operations Research vol 37no 6 pp 865ndash892 1989

[10] T N Bui and B R Moon ldquoGenetic algorithm and graphpartitioningrdquo IEEETransactions on Computers vol 45 no 7 pp841ndash855 1996

[11] R Battiti and A A Bertossi ldquoGreedy prohibition and reactiveheuristics for graph partitioningrdquo IEEE Transactions on Com-puters vol 48 no 4 pp 361ndash385 1999

[12] E Rolland H Pirkul and F Glover ldquoTabu search for graphpartitioningrdquo Annals of Operations Research vol 63 no 2 pp209ndash232 1996

[13] T N Bui and L C Strite ldquoAn ant system algorithm for graphbisectionrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference pp 43ndash51 July 2002

[14] I Hwang Y-H Kim and B-R Moon ldquoMulti-attractor genereordering for graph bisectionrdquo inProceedings of theGenetic andEvolutionary Computation Conference pp 1209ndash1215 July 2006

[15] Y-H Kim and B-R Moon ldquoLock gain based graph partition-ingrdquo Journal of Heuristics vol 10 no 1 pp 37ndash57 2004

[16] P Merz and B Freisleben ldquoFitness landscapes memetic algo-rithms and greedy operators for graph bipartitioningrdquo Evolu-tionary Computation vol 8 no 1 pp 61ndash91 2000

[17] HMuhlenbein and TMahnig ldquoEvolutionary optimization andthe estimation of search distributions with applications to graph

10 Mathematical Problems in Engineering

bipartitioningrdquo International Journal of Approximate Reasoningvol 31 no 3 pp 157ndash192 2002

[18] A G Steenbeek E Marchiori and A E Eiben ldquoFindingbalanced graph bi-partitions using a hybrid genetic algorithmrdquoin Proceedings of the IEEE International Conference on Evolu-tionary Computation pp 90ndash95 May 1998

[19] Y Yoon and Y-H Kim ldquoVertex ordering clustering andtheir application to graph partitioningrdquoApplied Mathematics ampInformation Sciences vol 8 no 1 pp 135ndash138 2014

[20] S-J Kang and B-R Moon ldquoA hybrid genetic algorithm formultiway graph partitioningrdquo in Proceedings of the Genetic andEvolutionary Computation Conference pp 159ndash166 July 2000

[21] J-P Kim and B-R Moon ldquoA hybrid genetic search for multi-way graph partitioning based on direct partitioningrdquo in Pro-ceedings of the Genetic and Evolutionary Computation Confer-ence pp 408ndash415 July 2001

[22] Y-H Kim A Moraglio Y Yoon and B-R Moon ldquoGeometriccrossover for multiway graph partitioningrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1217ndash1224 July 2006

[23] A Moraglio Y-H Kim Y Yoon and B-R Moon ldquoGeometriccrossovers for multiway graph partitioningrdquo Evolutionary Com-putation vol 15 no 4 pp 445ndash474 2007

[24] AMoraglio Y-H Kim Y Yoon B-RMoon and R Poli ldquoGeo-metric crossover for permutations with repetitions applicationto graph partitioningrdquo in Proceedings of the PPSN Workshopon Evolutionary AlgorithmsmdashBridging Theory and PracticeSeptember 2006

[25] A J Soper C Walshaw and M Cross ldquoA combined evolu-tionary search and multilevel optimisation approach to graph-partitioningrdquo Journal of Global Optimization vol 29 no 2 pp225ndash241 2004

[26] Y Yoon Y-H Kim A Moraglio and B-R Moon ldquoQuotientgeometric crossovers and redundant encodingsrdquo TheoreticalComputer Science vol 425 pp 4ndash16 2012

[27] J Kim I Hwang Y-H Kim and B-R Moon ldquoGeneticapproaches for graph partitioning a surveyrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 473ndash480 July 2011

[28] S Dutt and W Deng ldquoA probability-based approach to VLSIcircuit partitioningrdquo in Proceedings of the Design AutomationConference pp 100ndash105 June 1996

[29] B Krishnamurthy ldquoAn improved min-cut algorithm for parti-tioning VLSI networksrdquo IEEE Transactions on Computers vol33 no 5 pp 438ndash446 1984

[30] L A Sanchis ldquoMultiple-way network partitioningrdquo IEEETrans-actions on Computers vol 38 no 1 pp 62ndash81 1989

[31] G Laszewski ldquoIntelligent structural operators for the k-waygraph partitioning problemrdquo in Proceedings of the InternationalConference on Genetic Algorithms pp 45ndash52 July 1991

[32] H Muhlenbein ldquoParallel genetic algorithms in combinatorialoptimizationrdquo in Computer Science and Operations Research OBalci R Sharda and S Zenios Eds pp 441ndash453 PergamonOxford UK 1992

[33] S-S Choi and B-R Moon ldquoNormalization in genetic algo-rithmsrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference vol 2723 of Lecture Notes in ComputerScience pp 862ndash873 2003

[34] S-H Kim Y-H Kim and B-R Moon ldquoA hybrid geneticalgorithm for themax cut problemrdquo inProceedings of theGeneticand Evolutionary Computation Conference pp 416ndash423 July2001

[35] Y-H Kim Y-K Kwon and B-R Moon ldquoProblem-independ-ent schema synthesis for genetic algorithmsrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1112ndash1122 July 2003

[36] K Seo S Hyun and Y-H Kim ldquoAn edge-set representationbased on spanning tree for searching cut spacerdquo IEEE Trans-actions on Evolutionary Computation 2014

[37] Y-H Kim Combinatorial optimization based on effective localsearch genetic algorithms and problem space analyses [PhDthesis] Seoul National University Seoul Republic of Korea2005

[38] Y-H Kim ldquoAn enzyme-inspired approach to surmount barriersin graph bisectionrdquo in Proceedings of the International Confer-ence on Computational Science and Its Applications vol 5072 ofLecture Notes in Computer Science pp 841ndash851 2008

[39] S Dutt and W Deng ldquoVLSI circuit partitioning by cluster-removal using iterative improvement techniquesrdquo in Proceed-ings of the IEEEACM International Conference on Computer-Aided Design pp 194ndash200 November 1996

[40] S Dutt and W Deng ldquoCluster-aware iterative improvementtechniques for partitioning large VLSI circuitsrdquo ACM Transac-tions on Design Automation of Electronic Systems vol 7 no 1pp 91ndash121 2002

[41] Y Yoon and Y-H Kim ldquoAn efficient genetic algorithm formaximum coverage deployment in wireless sensor networksrdquoIEEE Transactions on Cybernetics vol 43 no 5 pp 1473ndash14832013

[42] Y Yoon Y-H Kim AMoraglio and B-RMoon ldquoA theoreticaland empirical study on unbiased boundary-extended crossoverfor real-valued representationrdquo Information Sciences vol 183no 1 pp 48ndash65 2012

[43] BMonien andRDiekmann ldquoA local graph partitioning heuris-tic meeting bisection boundsrdquo in Proceedings of the SIAM Con-ference on Parallel Processing for Scientific Computing March1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

bipartitioningrdquo International Journal of Approximate Reasoningvol 31 no 3 pp 157ndash192 2002

[18] A G Steenbeek E Marchiori and A E Eiben ldquoFindingbalanced graph bi-partitions using a hybrid genetic algorithmrdquoin Proceedings of the IEEE International Conference on Evolu-tionary Computation pp 90ndash95 May 1998

[19] Y Yoon and Y-H Kim ldquoVertex ordering clustering andtheir application to graph partitioningrdquoApplied Mathematics ampInformation Sciences vol 8 no 1 pp 135ndash138 2014

[20] S-J Kang and B-R Moon ldquoA hybrid genetic algorithm formultiway graph partitioningrdquo in Proceedings of the Genetic andEvolutionary Computation Conference pp 159ndash166 July 2000

[21] J-P Kim and B-R Moon ldquoA hybrid genetic search for multi-way graph partitioning based on direct partitioningrdquo in Pro-ceedings of the Genetic and Evolutionary Computation Confer-ence pp 408ndash415 July 2001

[22] Y-H Kim A Moraglio Y Yoon and B-R Moon ldquoGeometriccrossover for multiway graph partitioningrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1217ndash1224 July 2006

[23] A Moraglio Y-H Kim Y Yoon and B-R Moon ldquoGeometriccrossovers for multiway graph partitioningrdquo Evolutionary Com-putation vol 15 no 4 pp 445ndash474 2007

[24] AMoraglio Y-H Kim Y Yoon B-RMoon and R Poli ldquoGeo-metric crossover for permutations with repetitions applicationto graph partitioningrdquo in Proceedings of the PPSN Workshopon Evolutionary AlgorithmsmdashBridging Theory and PracticeSeptember 2006

[25] A J Soper C Walshaw and M Cross ldquoA combined evolu-tionary search and multilevel optimisation approach to graph-partitioningrdquo Journal of Global Optimization vol 29 no 2 pp225ndash241 2004

[26] Y Yoon Y-H Kim A Moraglio and B-R Moon ldquoQuotientgeometric crossovers and redundant encodingsrdquo TheoreticalComputer Science vol 425 pp 4ndash16 2012

[27] J Kim I Hwang Y-H Kim and B-R Moon ldquoGeneticapproaches for graph partitioning a surveyrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 473ndash480 July 2011

[28] S Dutt and W Deng ldquoA probability-based approach to VLSIcircuit partitioningrdquo in Proceedings of the Design AutomationConference pp 100ndash105 June 1996

[29] B Krishnamurthy ldquoAn improved min-cut algorithm for parti-tioning VLSI networksrdquo IEEE Transactions on Computers vol33 no 5 pp 438ndash446 1984

[30] L A Sanchis ldquoMultiple-way network partitioningrdquo IEEETrans-actions on Computers vol 38 no 1 pp 62ndash81 1989

[31] G Laszewski ldquoIntelligent structural operators for the k-waygraph partitioning problemrdquo in Proceedings of the InternationalConference on Genetic Algorithms pp 45ndash52 July 1991

[32] H Muhlenbein ldquoParallel genetic algorithms in combinatorialoptimizationrdquo in Computer Science and Operations Research OBalci R Sharda and S Zenios Eds pp 441ndash453 PergamonOxford UK 1992

[33] S-S Choi and B-R Moon ldquoNormalization in genetic algo-rithmsrdquo in Proceedings of the Genetic and Evolutionary Com-putation Conference vol 2723 of Lecture Notes in ComputerScience pp 862ndash873 2003

[34] S-H Kim Y-H Kim and B-R Moon ldquoA hybrid geneticalgorithm for themax cut problemrdquo inProceedings of theGeneticand Evolutionary Computation Conference pp 416ndash423 July2001

[35] Y-H Kim Y-K Kwon and B-R Moon ldquoProblem-independ-ent schema synthesis for genetic algorithmsrdquo in Proceedings ofthe Genetic and Evolutionary Computation Conference pp 1112ndash1122 July 2003

[36] K Seo S Hyun and Y-H Kim ldquoAn edge-set representationbased on spanning tree for searching cut spacerdquo IEEE Trans-actions on Evolutionary Computation 2014

[37] Y-H Kim Combinatorial optimization based on effective localsearch genetic algorithms and problem space analyses [PhDthesis] Seoul National University Seoul Republic of Korea2005

[38] Y-H Kim ldquoAn enzyme-inspired approach to surmount barriersin graph bisectionrdquo in Proceedings of the International Confer-ence on Computational Science and Its Applications vol 5072 ofLecture Notes in Computer Science pp 841ndash851 2008

[39] S Dutt and W Deng ldquoVLSI circuit partitioning by cluster-removal using iterative improvement techniquesrdquo in Proceed-ings of the IEEEACM International Conference on Computer-Aided Design pp 194ndash200 November 1996

[40] S Dutt and W Deng ldquoCluster-aware iterative improvementtechniques for partitioning large VLSI circuitsrdquo ACM Transac-tions on Design Automation of Electronic Systems vol 7 no 1pp 91ndash121 2002

[41] Y Yoon and Y-H Kim ldquoAn efficient genetic algorithm formaximum coverage deployment in wireless sensor networksrdquoIEEE Transactions on Cybernetics vol 43 no 5 pp 1473ndash14832013

[42] Y Yoon Y-H Kim AMoraglio and B-RMoon ldquoA theoreticaland empirical study on unbiased boundary-extended crossoverfor real-valued representationrdquo Information Sciences vol 183no 1 pp 48ndash65 2012

[43] BMonien andRDiekmann ldquoA local graph partitioning heuris-tic meeting bisection boundsrdquo in Proceedings of the SIAM Con-ference on Parallel Processing for Scientific Computing March1997

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of