Solving Steiner Tree Problems in Graphs to · PDF fileSolving Steiner Tree Problems in Graphs to Optimality ... on an integer programming formulation ... ), SOLVING STEINER TREE PROBLEMS

Solving Steiner Tree Problems in Graphsto Optimality

T. Koch, A. Martin

Konrad-Zuse-Zentrum fur Informationstechnik Berlin, Takustrasse 7, D-14195 Berlin–Dahlem,Germany

Received 11 February 1997; accepted 9 April 1998

Abstract: In this paper, we present the implementation of a branch-and-cut algorithm for solving Steinertree problems in graphs. Our algorithm is based on an integer programming formulation for directedgraphs and comprises preprocessing, separation algorithms, and primal heuristics. We are able to solvenearly all problem instances discussed in the literature to optimality, including one problem that—to ourknowledge—has not yet been solved. We also report on our computational experiences with some verylarge Steiner tree problems arising from the design of electronic circuits. All test problems are gatheredin a newly introduced library called SteinLib that is accessible via the World Wide Web. q 1998 JohnWiley & Sons, Inc. Networks 32: 207–232, 1998

Keywords: branch-and-cut; cutting planes; reduction methods; Steiner tree; Steiner tree library

1. INTRODUCTION are exact algorithms, heuristic procedures, approximationalgorithms, polynomial algorithms for special instances,polyhedral approaches, preprocessing techniques, andGiven an undirected graph G Å (V, E) and a node set Tmore. Excellent surveys were given in Winter [40], Mac-⊆ V, a Steiner tree for T in G is a subset S ⊆ E of theulan [32], Hwang and Richards [25], and Hwang et al.edges such that (V ( S) , S) contains a path from s to t for[26]. To solve the Steiner tree problem to optimality,all s , t √ T , where V ( S) denotes the set of nodes incidentAneja [1] proposed a row-generation algorithm based onto an edge in S . In other words, a Steiner tree is an edgean undirected formulation, Dreyfus and Wagner [11] andset S that spans T . The Steiner tree problem is to find aLawler [29] used dynamic programming techniques,minimal Steiner tree with respect to some given edge costsBeasley [4, 5] presented a Lagrangean relaxation ap-ce , e √ E . This problem is known to be NP-hard (Karpproach, Wong [43] described a dual-ascent method, Lu-[28]), even for grid graphs (Garey and Johnson [18]).cena [31] combined Lagrangean and polyhedral methods,Nourished from the increasing demand in the designand Chopra et al. [8] developed a branch-and-cut algo-of electronic circuits, the solution of Steiner tree problemsrithm. In particular, polyhedral methods have turned out

has received considerable and strongly growing attentionto be quite powerful in finding optimal solutions for vari-

in the last 20 years. Among the proposed solution methodsous Steiner tree problems. Reasons for that are the betterunderstanding of the associated polyhedra, the availabilityof fast and robust LP solvers, and the experience gainedCorrespondence to: A. Martinto turn the theory into an algorithmic tool.Mathematical Subject Classification (1995): 05C40, 90C06, 90C10,

90C35 This paper moves within this framework and presents

q 1998 John Wiley & Sons, Inc. CCC 0028-3045/98/030207-26

207

836/ 8u26$$0836 08-17-98 14:34:18 netwa W: Networks

208 KOCH AND MARTIN

(uSP)

a branch-and-cut algorithm. It is strongly related to thealgorithm described in Chopra et al. [8]; we solve thesame integer programming formulation, again by meansof the separation of cutting planes. However, the newalgorithm differs considerably, not only in several aspects

min cTx

( i) x(d(W )) ¢ 1, for all W , V,

W > T x M,

(V "W ) > T x M,

( ii) 0 ° xe ° 1, for all e √ E ,

( iii) x integer,

of implementation but also due to some extensions. Themain extensions are a more effective preprocessing phaseby incorporating three preprocessing tests, an extensionof the initial integer program with so-called flow-balanceconstraints, and a more careful and more efficient separa-tion of active cut constraints resulting in leaner LPs. In where d(X ) denotes the cut induced by X ⊆ V, that is,Section 2, we review two different integer programming the set of edges with one end node in X and one in itsformulations. The second, on which the branch-and-cut complement, and x(F) :Å (e√F xe , for F ⊆ E . It is easyalgorithm is based, is a bidirected version of the first. In to see that there is a one-to-one correspondence betweenSection 3, we discuss preprocessing and exploit ideas Steiner trees in G and 0/1 vectors satisfying (uSP) (i) .known from the literature. In particular, our presolve algo- Hence, the Steiner tree problem can be solved via (uSP).rithm includes three strong reduction techniques of Duin Another way to model the Steiner tree problem is toand Volgenant [13, 14]. Our computational results dem- consider the problem in a directed graph. We replace eachonstrate how important preprocessing is: Without this edge [u , £] √ E by two antiparallel arcs (u , £) and (£,tool, it would not have been possible to solve any of the u) . Let A denote this set of arcs and D Å (V, A) , thelarge instances. Details of the cutting plane phase of our resulting digraph. We choose some terminal r √ T , whichbranch-and-cut algorithms are discussed in Section 4. It will be called the root. A Steiner arborescence (rootedincludes refined separation strategies (resulting in leaner at r) is a set of arcs S ⊆ A such that (V ( S) , S) containsLPs) and improved primal heuristics such that at an ear- a directed path from r to t for all t √ T" {r}. Obviously,lier stage the lower- and upper-bound values meet. Exten- there is a one-to-one correspondence between (undi-sive tests are given in Section 5. We solve almost all test rected) Steiner trees in G and Steiner arborescences in Dinstances from the literature including one problem that— which contain at most one of two antiparallel arcs. Thus,to our knowledge—has not yet been solved and find the if we choose arc costs c

u (u,£ ) :Å cu (£,u ) :Å c[u,£ ] , for [u , £]

optimal solution for many very large instances arising √ E , the Steiner tree problem can be solved by findingfrom real-world problems in the design of electronic cir- a minimal Steiner arborescence with respect to c

u. Note

cuits. We introduce a library for Steiner tree problems that there is always an optimal Steiner arborescence whichcalled SteinLib ( including most of the models from the does not contain an arc and its antiparallel counterpart,literature and all new VLSI-instances discussed in this since c

u¢ 0. Introducing variables ya for a √ A with the

paper) . This library is available via anonymous ftp or interpretation that ya :Å 1, if arc a is in the Steiner arbores-from WWW at URL: ftp:// ftp.zib.de/pub/Packages/mp- cence, and ya :Å 0, otherwise, we obtain the integer pro-testdata/steinlib/ . gram

(dSP)

2. INTEGER PROGRAMMINGFORMULATION

min cu

Ty

( i) y(d/(W )) ¢ 1, for all W , V,

r √ W ,

(V "W ) > T x M,

( ii) 0 ° ya ° 1, for all a √ A ,

( iii) y integer,

In this section, we present the integer programming for-mulation that we are going to solve with our branch-and-cut algorithm. Let an undirected graph G Å (V, E) withedge costs ce¢ 0, e√ E , be given. We assume throughoutthe paper that the edge costs are nonnegative and integer. where d/(X ) :Å {(u , £) √ AÉu √ X , £ √ V "X } for X

, V, that is, the set of arcs with tail in X and head in itsIn addition, there is a node set T ⊆ V, called the set ofterminals. We will denote an instance of the Steiner tree complement. Again, it is easy to see that each 0/1 vector

satisfying (dSP) (i) corresponds to a Steiner arbores-problem by the triple ST(G , T , c) .A canonical way to formulate the Steiner tree problem cence, and, conversely, the incidence vector of each

Steiner arborescence satisfies (dSP) (i) – (iii ) . How areas an integer program is to introduce, for each edge e√ E , a variable xe , indicating whether e is in the Steiner the models (uSP) and (dSP) related?

Polyhedral aspects of both models are intensively dis-tree (xeÅ 1) or not (xeÅ 0). Consider the integer program


SOLVING STEINER TREE PROBLEMS IN GRAPHS TO OPTIMALITY 209

large scale. The idea, in general, is to detect unnecessaryinformation in the problem description and to reduce thesize of the problem by logical implications. For theSteiner tree problem, many reduction methods are dis-cussed in the literature and have been shown to be veryeffective for solving large instances; see, for example,Balakrishnan and Patel [2] , Beasley [4] , Chopra et al.[8] , Duin [12], Duin and Volgenant [14], Lucena [31],Winter [41], and Winter and Smith [42]. These methodsfocus on detecting special configurations that allow oneto neglect certain edges and/or nodes for the optimizationor they show that some edges and/or nodes are containedin some optimal solution. In this section, we sketch themain concepts from the literature and show how they areincorporated in our code.Fig. 1. Original problem.

How successful preprocessing methods might be inreducing the size of some problem is demonstrated in

cussed in the literature. The undirected model was studied Figures 1 and 2. Figure 1 shows the original graph ofin Grotschel and Monma [23], Goemans [19], Goemans problem br (complete graph on 58 nodes; for a descrip-and Myung [20], and Chopra and Rao [9, 10], whereas tion of the problem, see Section 5), and Figure 2, thethe directed version, in Ball et al. [3] , Fischetti [17], graph that we obtain after applying our preprocessingGoemans and Myung [20], and Chopra and Rao [9, 10]. algorithm.Chopra and Rao [9] and Goemans and Myung [20] re-lated both formulations. Chopra and Rao [9] showed that

3.1. Degree-Test Ithe optimal value of the LP relaxation of the directedThe following tests summarized under the name degree-model zd :Å min{c

u

TyÉy satisfies (dSP) (i) and (ii)} istest I (see [4]) are easy to check:greater or equal to the corresponding value of the undi-

rected formulation zu :Å min{cTxÉx satisfies (uSP) (i)( i ) A nonterminal node of degree one can be removed.and (ii)}. Even, if the undirected formulation is tightened

(ii) If a nonterminal node £ is of degree two, node £ andby the so-called Steiner partition inequalities (seethe two incident edges [u , £] and [£, w] , u x w ,Grotschel and Monma [23]; Chopra and Rao [9]) andcan be replaced by an edge connecting u and w ofodd hole inequalities (see Chopra and Rao [9]) , thiscost c[u,w ] Å c[u,£ ] / c[£,w ] .relation holds. In addition, Goemans and Myung [20]

(iii ) An edge incident to a terminal node of degree oneshowed that zd is independent of the choice of the root r .is always in an optimal solution.These results suggest the directed model and we followed

this suggestion. Nevertheless, one disadvantage of the (iv) If an edge e is of minimal cost among the edgesdirected model is that the number of variables is doubled. incident to a terminal node, and the other end nodeBut it will turn out that this is not really a bottleneck, is also a terminal, then e is choosable in any optimalsince we are minimizing a nonnegative objective function, solution.and thus the variable of one of two antiparallel arcs willusually be at its lower bound. 3.2. Special-Distance-Test

It should be mentioned that further models to solveThis test ( introduced in Duin and Volgenant [13]) com-the Steiner tree problem can be found in the literature;putes for each pair of nodes a number (called the specialfor example, models based on flow formulations (Wongdistance) which can be exploited to remove some edges.[43]; Maculan [32]) or models extending the undirected

formulation by introducing node variables (Lucena [31];Goemans and Myung [20]) . Relations between relax-ations of these and the above-discussed models can befound in Wong [43], Maculan [32], Duin [12], andGoemans and Myung [20].

3. PREPROCESSING

Preprocessing is a very important algorithmic tool in solv-Fig. 2. Reduced problem.ing combinatorial and integer programming problems of


210 KOCH AND MARTIN

Definition 3.1 (Special Distance). Let two nodes u , £ has to be introduced. (In case of parallel edges, only oneedge will be retained.) Of course, this might create many√ V with u x £ be given, and consider some path P ⊆ E

connecting u and £. Set TP Å V ( P) > T < {u , £} and new edges, but, in general, most of these can be elimi-nated by the special-distance-test.let

The running time for this test is O(2mrg) , where g

denotes the time for computing a minimal spanning tree.b(P) Å max{c(F)ÉF ⊆ P is a path connectingDue to the exponential behavior, we perform this test

two nodes from TP such that only for m ° 3. In fact, the bottleneck degree m testgeneralizes the ones in Section 3.1 (i) , where m Å 1, andÉTP > V ( F)É Å 2}.Section 3.1 (ii ) , where m Å 2.

The number

3.4. Terminal-Distance-Tests(u , £) Å min{b(P)ÉP is a path connecting u and £}

In this test, we consider a connected subgraph H Å (W ,is called the special distance (between u and £) . F) of G with T > W x M and T"W x M. Let e

To give an idea what s(u , £) means, consider each Å argmine =√d (W )ce = and f Å argminf=√d (W )" {e }cf= be aterminal as a petrol station and suppose you want to drive shortest and a second shortest edge of the cut induced by W.from location u to £. Then, s(u , £) denotes the distance Edge e Å [u , £] with u √ W and £ √ V "W is part ofyou must be able to drive without refilling if you choose some optimal solution of ST(G , T , c) and can thus beamong all possible routes. Note that the following rela- contracted, iftions

cf ¢ du / ce / d£,

s(u , £) ° d(u , £) ° c[u,£ ]

with du Å min{d( t , u)Ét √ T > W } and d£Å min{d( t ,

hold, where d (u , £ ) denotes the length of a shortest£)Ét √ T"W }.

path between u and £. The special distance can be com-Duin and Volgenant [14] introduced this test and gave

puted by a modified shortest path algorithm (cf. Hwangan implementation in O(ÉVÉ

3) steps. In Duin [12], it iset al. [ 26 ] ) .

shown that the detection of all edges satisfying the condi-Given the values s(u , £) for all u , £ √ V, there is an

tion of the terminal-distance-test needs only O(ÉVÉ2)

easy and very effective test for deleting edges. An optimalsteps. Note that the last two tests in Section 3.1 (iii ) and

solution S* of a Steiner tree problem ST(G , T , c) cannot( iv) are special cases of the terminal-distance-test. Two

contain any edge [u , £] √ E with s(u , £) õ c[u,£ ] . other special cases are the R-R aggregation method ofThe special-distance-test is a generalization of many

Balakrishnan and Patel [2] and the nearest vertex test ofother tests known in the literature; this was comprehen-

Beasley [4] .sively treated in Duin and Volgenant [13]. Concerningimplementation, it should be noted that certain specialcases of this test can be implemented more efficiently. 3.5. ResultsHowever, one can also resort to a well-performing ap-

When it comes to implement these reduction methods,proximation of the special-distance-test that runs inseveral questions arise: Which of these tests should beO(ÉVÉ log ÉVÉ / ÉEÉ / ÉTÉ2) (cf. Duin [12]) .implemented? For each single test, should all cases bechecked (complete test) which might result in high run-

3.3. Bottleneck Degree m Test ning times or should one restrict the search to certainpromising special cases which might result in an incom-The bottleneck degree m test introduced in Duin and Vol-plete test? In which order should the methods be called?genant [14] is the following: Consider some node £ √ NHow often should they be called? Some reduction of onewith Éd(£)É Å m . Let (W , F) be the complete graph ontest might give rise to further reductions by some othernode set W :Å V (d(£))" {£} with edge costs s

V [u,£ ] Å s(u ,(already performed) test. These questions were already

£) for [u, £] √ F. If, for all subsets U ⊆ W with ÉUÉ ¢ 3,addressed in Duin and Volgenant [14]. With respect toour algorithm, we should also answer the questions: HowsV (B*) ° ∑

u√U

c[£,u ] ,much effort and computation time should one spent inthe preprocessing phase? At what point is it usually betterto switch over to the branch-and-cut phase? We tried towhere B* is the edge set of a minimal spanning tree in

(W , F) , holds, node £ can be deleted, and for all u , w find answers in the following way: First, we implementedall the tests and each test in the complete version. We√ W , u x w , edge [u , w] with cost c[u,w ] Å c[u,£ ] / c[£,w ]



called all these tests consecutively and iterated this pro- (9) branch if necessary, otherwise remove the leaffrom the treecess until no more reductions could be found. Of course,

this might be very time consuming but for large difficult (10) until branch-and-bound tree is empty(11) print the optimal solutionproblems it might be worth to reduce as much as possible

(see Section 5). For small and medium-sized problems, (12) STOP.the situation is different. Often it did not pay to performa complete test, but rather to switch to the branch-and- In the Initialization phase, we set up the first LP and

initialize the branch-and-bound tree with the root nodecut phase which usually solved the (reduced) problemvery fast. We performed many test runs to find a balance representing the whole problem. In our case, the starting

LP is essentially empty, consisting only of the trivialbetween the total running times and the success of thereduction methods. Algorithm 3.2 shows our default se- inequalities (dSP) (ii ) . We experimented with initial cuts

for the first LP by doing a breadth-first search from thelection:root to every other terminal and adding the cuts betweennodes of different depth. Although these cuts have disjointAlgorithm 3.2 (Default Presolve)support for each root-terminal pair, only the smaller in-stances profited from this idea. While the number of cut-(1) Degree-Test Iting plane iterations [i.e., the number of runs through(2) Special-Distance-TestSteps (4) – (8)] needed to solve the problems was always(3) Degree-Test Ismaller, the effect from initially having a lot of dense(4) Terminal-Distance-Testinequalities ( i.e., inequalities with many nonzero entries)(5) Special-Distance-Testin the LP considerably slows down the whole process.(6) Degree-Test I

For solving the linear programs, we used CPLEX*,(7) Special-Distance-TestVersions 4.0.9 and 5.0, a very fast and robust linear pro-(8) Degree-Test Igramming solver, which features both a primal and dual(9) Returnsimplex solver and a primal-dual barrier solver. We usedthe dual simplex algorithm, since the LPs from one itera-Note that the bottleneck degree m test is not includedtion to the next stay dual feasible, when cutting planesin our default strategy. For some difficult instances, how-are added or variables are fixed to one of their bounds.ever, it pays to use the bottleneck degree m test and iterateIt turned out that the best pricing strategy was steepest-all four tests as along as there is some reduction possible.edge pricing, that is, to select a variable entering the basisThe success of our presolve strategy is illustrated in Sec-that has largest (obtuse) angle with the gradient of thetion 5.objective function. However, for some instances (in par-ticular, for large grid problems), the arising LPs arehighly primal and dual-degenerated.

4. IMPLEMENTATION DETAILS We tried to avoid degeneracy by perturbing theobjective function. We used c Å c

u0 b1a , where

In this section, we describe the implementation of ourb Å minS 1001 ,

12 (ÉAÉ / 1 ) D , and 1a √ [0, 1) isbranch-and-cut algorithm for solving the Steiner tree

problem. We assume that the reader is familiar with thesome uniformly distributed random number for each ageneral outline of a branch-and-cut algorithm (see Ca- √ A . Since we assumed that c is an integer, our choiceprara and Fischetti [7] for a survey). Algorithm 4.1 pres-of c ensures that an optimal solution with c is also optimalents the main steps of such an algorithm:for cu . The running times for solving the LPs were alwaysbetter with the perturbed objective function than with the

Algorithm 4.1 (Branch-and-Cut Algorithm)original. Nevertheless, some of the larger problems con-tinued to show signs of degeneracy. We tried two further

(1) Initialization ways to remedy degeneracy: First, we perturbed the objec-(2) repeat tive function with b Å 0.1. This, however, requires reopti-(3) select a leaf from the tree and consider the associ- mization with the original objective function after the

ated LP problem has been solved for the perturbed objective func-(4) repeat ( iterate) tion. Sometimes, this reoptimization step needed several(5) solve the LP thousand simplex iterations and we obtained a significant(6) call primal heuristics speed up only in very few cases. Second, we tried the(7) separate violated inequalities and add them to

the LP(8) until there are no violated inequalities * CPLEX is a registered trademark of ILOG.


212 KOCH AND MARTIN

primal-dual barrier solver of the CPLEX package. The heuristic is called in our code is specified by some param-eter (default is every five cutting plane iterations) .barrier code does not suffer from degeneracy, but has to

solve each LP from scratch so that, on average, it could In 138 out of 414 test examples, the first call to theheuristic found the optimal solution, and in 90% of thenot outperform the dual simplex method with our initial

perturbation. Thus, our default choice to solve the LPs is cases, the gap [(heuristic solution 0 lower bound)/ lowerbound] was below 5%.to use the perturbed objective function c and apply the

dual simplex algorithm with steepest-edge pricing. We also experimented with the Rayward-Smith [35]heuristic. The results are quite promising; however, amain bottleneck is the running time, especially for big4.1. Branching and Selecting Leafsproblems. The reason is that the heuristic requires all-to-all node distances, and due to memory limitations, weIn Step (9), if it is necessary to branch, we use strong

branching (Bixby [6]) , that is, we determine a set of must compute these on the fly, so most of the time isspent for calculating shortest paths.variables whose LP value is close to 0.5, perform for each

variable of this set a certain number of simplex iterationsfor the linear program where the variable is set to one orzero, and, finally, select the variable in the set as

4.3. Separating Inequalitiesbranching variable that obtained the best increase in theLP value. In this section, we start with the description of our separa-

We run through the branching tree in a depth-first- tion routines for the cut inequalities (dSP) (i) . We firstsearch fashion. The reasons are that the memory require- discuss how the generic cut separation works and give anments for the whole tree stay small and that we try to efficient implementation. In the following three subsec-find a good primal solution as soon as possible. It almost tions, we discuss three improvements on the generic cuts:never happened that our branching tree grew to much. back cuts, nested cuts, and creep-flow cuts. All these cutsBranching was a rather rare event in our computations aim at selecting violated cuts that give the most progressanyway (within the time limit and with the default param- in terms of an increase in the lower bound with respecteter setting branching was necessary only in 37 of 414 to the running time. We finally present a further class ofcases; see Section 5). inequalities, the so-called flow-balance inequalities, that

may strengthen the LP relaxation further. All four sugges-tions can be combined with each other, resulting in 164.2. Primal Heuristicpossible ways to separate inequalities. Based on some

The primal heuristic that we use is basically the one intro- computational tests in the last subsection, we present ourduced by Takahashi and Matsuyama [38]. The idea of final separation strategy.this heuristic is to start from one terminal and connect aterminal by a shortest path that is closest to the startingterminal. The next terminal is chosen among the re- 4.3.1. Generic Cutsmaining terminals in such a way that it is closest to thealready existing path or subtree in general. This process It is well known that the separation problem for the cut

inequalities (dSP) (i) can be solved by any max-flowis continued until all terminals are connected. As edgecosts for this heuristic we use (1 0 xe)rce for e √ E , if algorithm and can thus be solved in polynomial time. We

regard the LP solution as capacities in the graph andx is the optimal solution of the current LP, that is, we tryto prefer those edges that are already chosen in the LP check, for each t √ T" {r}, whether the minimal (r , t)-

cut is less than one. If so, a violated cut inequality issolution. (A slightly different objective function was sug-gested in Lucena [31] who used as edge costs 0, if xe found; otherwise, there is none. We add inequalities only

if they are violated by at least some epsilon (default isÅ 1, and ce , otherwise; Chopra et al. [8] used the originaledge costs, but considered only edges e with positive LP 1004) . To determine a minimal (r , t)-cut, for all t

√ T" {r}, we must call, in principle, (ÉTÉ 0 1) times avalue xe ú 0.) As suggested in Rayward-Smith and Clare[36], we also try to improve the heuristic solution by max-flow algorithm. However, Hao and Orlin [24]

showed that by a careful implementation of a preflow-computing a minimal spanning tree among the chosennodes and prune nonterminal leaves as long as possible. push algorithm the time to determine minimal cuts from

the root node r to all other nodes is comparable with theA parameter to be specified for this heuristic is thestarting terminal. Since running the algorithm for all ter- time to find a single (r , t)-cut. If we use the highest label

preflow-push algorithm of Goldberg and Tarjan [21] theminals is usually too time-consuming, we made the fol-lowing selection: We always try the terminal which gave overall running time of the Hao–Orlin algorithm to deter-

mine a minimal (r , t)-cut, for all t √ T" {r}, isthe best solution so far and try in addition up to 10 ran-domly selected terminals. The frequency in which the O(ÉVÉ

2ÉEÉ1/2 ) .



ered, the time needed for reoptimization the linear pro-grams decreased by a factor of 10. Moreover, the reduc-tion in the number of cutting plane iterations by usingthese ideas over just adding pure (r , t)-cuts is betweentwo and three orders of magnitude.

4.3.5. Flow-Balance Inequalities

In our cutting plane phase we take another class of ine-qualities into consideration. An (optimal) Steiner arbores-cence can be viewed as a set of flows sending one unit

Fig. 3. alue7229. from the root r to each terminal in T" {r}. This meansthat for all nonterminal nodes that are not branching nodesin the Steiner arborescence the flow-balance equality

4.3.2. Back Cuts y(d0(£))Å y(d/(£)) must hold, and for the other nonter-minal nodes, y(d0(£)) ° y(d/(£)) . This is expressed inChopra et al. [8] described a method to increase the num-the following set of inequalities:ber of separated inequalities by swapping the flow on

each arc and checking in addition all ( t , r)-cuts, for t√ T" {r}. A drawback here is that we cannot use thespeed-up feature mentioned above, since for each ( t , r)-cut computation, the source node changes and thus thealgorithm has to start from scratch again. However, as

Å 0, if £ Å r ;

y(d0(£)) Å 1, if £ √ T" {r} ;

° 1, if £ √ N ;

y(d0(£)) ° y(d/(£)) , for £ √ N ;

y(d0(£)) ¢ ye , for all e √ d/(£) ,

£ √ N .

we will see in Section 4.3.6, this idea significantly im-proves the overall running time compared with the genericcut generation.

(1)

4.3.3. Nested Cuts

Another way to increase the number of violated inequali- Note that this system of inequalities is not valid for allties is to nest the cuts. After finding a minimal cut between Steiner arborescences (e.g., cycles are cut off) , but therer and some terminal t , we temporary fix all corresponding is always an optimal solution that satisfies these condi-variables in the actual LP solution to one and try again tions, since the objective function is nonnegative. Noteto find a cut between r and t . We repeat this procedure that the addition of inequalities (1) to (dSP) has alreadyuntil the flow between r and t is at least one. This idea been considered in Duin [12]. He gives an example wherecan be combined with back cuts so that we are trying to these inequalities strengthen the LP relaxation.find nested inequalities in both directions. The results ofthis procedure are usually an increase in the time spent 4.3.6. A Comparisonfor separation and reoptimization the linear programs per

We performed several tests to evaluate the performanceiteration, while the total number of cutting plane iterationsof these four separation routines and all its combinations.drastically decreases, resulting in a total running time ofFigures 3 and 4 show the results of all 16 possible separa-about one magnitude faster than without nested and backtion strategies for examples alue7229 and taq0631 (forcuts.

4.3.4. Creep-Flow

We obtained another major speedup in the performanceof our algorithm when we implemented the followingidea: Instead of trying to increase the number of separatedinequalities, we tried to raise the ‘‘quality’’ of the inequal-ities. Since most of the variables in our LP solution arezero, the optimal solution of the min-cut algorithm is notnecessarily arc minimal. So, we add a tiny capacity ofsome e ( in the code we use e Å 1006) to all arcs to getamong all weight minimal cuts one that is also arc mini-mal. While this increased the running time for computing

Fig. 4. taq0631.a minimal cut, since much more arcs have to be consid-


214 KOCH AND MARTIN

have chosen the ‘‘C--F’’ option as the final separationstrategy in our branch-and-cut algorithm.

4.4. Removing Inequalities

Sometimes in the iteration process inequalities becomenonbinding, that is, the slack of the inequalities are posi-tive. In these cases the inequality can be removed fromthe LP without changing its optimal value. Although theinequality can be violated again, it is, in general, a goodFig. 5. gr.idea to remove these inequalities in order to keep the LPsmall. To minimize the occurrences of these reviolatedinequalities, we added a ‘‘life’’ counter to each inequalitya description of these problems, see Section 5). F means

that flow-balance inequalities are applied; C , B , and N , currently in the LP. If the slack of an inequality is non-zero, the counter is decreased; if the slack is zero, it isindicates that creep-flow cuts, back cuts, and nested cuts

are added, respectively; and ‘‘----’’ indicates that just the reset to an initial value (in our implementation 5). If thecounter reaches zero, the inequality is removed. This waygeneric cut separation is applied. The x-axis shows the

16 possibilities sorted according to their total running we are delaying the removal of inequalities to a pointwhere it is more likely that it will never be used again.time in decreasing order from left to right. The curves

depict the total running time (in seconds) , the separationtime per iteration (in hundredth of seconds) , the total 4.5. Reduced Costs and Reduced Setnumber of added cuts ( in thousands) , and the average of Variablesnumber of nonzeros per row. We observe that the differ-ences in the running times are up to two orders of magni- Every time the primal heuristic finds a better solution, we

try to fix variables by the reduced-cost criterium. For atude (note that the y axis is logarithmically scaled). Wealso see that the strategy ‘‘-B--’’ that was used in Chopra discussion on reduced-cost fixing, see for instance Pad-

berg and Rinaldi [34]. With the exception of the classet al. [8] gives a significant speed up compared withjust adding generic cuts, although the separation time per of so-called incidence problems (see Section 5), this idea

has little effect on the performance of our algorithm. Dueiteration increases. This was already observed by Chopraet al. However, their strategy is not the overall best. For to the high degeneracy of the LPs, the reduced costs tend

to be very small and, thus, the reduced-cost criteriumalmost all combinations, it is better to apply flow-balanceinequalities. The same holds for creep-flows; the strategy (and possibly also other reduction methods that are based

on reduced costs, see Duin [12]) are likely to fail.with creep-flows is always better than the one without.The reason is mainly a significant reduction in the number Another commonly known idea is to work only on a

reduced set of variables by fixing variables temporally toof generated cuts. In both figures, the eight combinationsusing creep-flows together with ‘‘-BNF’’ are always the one of its bounds. After the problem has been solved on

the reduced set, we check the reduced costs of the tempo-best. We evaluated these nine strategies on some largerinstances. Figures 5 and 6 show the results for problem rally fixed variables, add them if necessary to the current

set of variables, and reoptimize. Instead of really remov-gr and msm1234 (note that the curves are not uniformlyand the y-axis is not logarithmically scaled any more to ing the variables that are fixed from the problem as it is

usually done in such a type of column-generation algo-better illustrate the differences of the strategies) .The ‘‘-BNF-’’ strategy shows a big increase in the rithm, we only fix these variables to their bounds and

keep them in the LP. CPLEX (the LP solver that we use)number of cuts and nonzeros resulting in high LP times(LP times are not shown in the diagram). The increasein the LP time per iteration is not completely compensatedby the decrease in the total number of iterations, resultingin running times that are not among the best. For largerinstances, this effect becomes even clearer. Again, werecognize the positive impact of the flow inequalities. Forexample, in problem gr , the ‘‘C--F’’ strategy has the bestnonzero per row index. In fact, this strategy is very robust:It is always among the four best, while the performanceof the other strategies does not seem to be predictable. Itis remarkable that the connection of C with B and N (with

Fig. 6. msm1234.or without F) does not outperform ‘‘C--F.’’ Therefore, we



manages fixed variables very efficiently so that we could difference between the lower and upper bound is less than1, the upper bound in the last column is shown in boldnot detect a major loss of performance (under the assump-

tion that limits of memory are not reached). The advan- face to indicate that the optimal solution was found. Ifthere is still a gap greater than 1 between LB and UB ,tage is that we do not have to take care of the management

of inequalities for which some of the variables are in the we have not found the optimal solution within the timelimit.current set of variables and some are not. For the limited

test runs that we performed for this column-generation Tables I and II show our results for the test seriesintroduced by Beasley [5] . Test set C is easy: We solveidea, we could not obtain a speedup on average.all instances with one exception within a minute. Interest-ing to note is that already the first call to the heuristic(without any dual information) gives in 11 out of 205. COMPUTATIONAL RESULTSexamples the optimal solution. Series D with 1000 nodesis a bit more difficult: The running times increase up toIn this section, we report on computational experiences6 minutes. However, the optimal solution is obtained inwith our branch-and-cut algorithm. Our code is imple-the root node in all but one case (d19) , that is, branchingmented in C and all runs (with the exception of the inci-was not necessary. To solve test series E (with the excep-dence problems, see the relevant page in the sequel) aretion of e18) , we need up to 2 hours per instance, althoughperformed on a Sun SPARC 20 Model 71. The test exam-still no branching is necessary. The number of cuts neededples include public available benchmarks discussed in theto solve these examples increases to about 66,000. Weliterature, some instances that authors of other Steinercould not detect a correlation between the number oftree codes made us available, and some realistic problemsviolated inequalities and the number of variables or termi-arising in the design of electronic circuits. All instancesnals. The number of inequalities in the final LP is ratherare gathered in the library SteinLib that is available viahigh compared with the number of cuts separated. Thisanonymous ftp or the World Wide Web.†means that the inequalities mostly stay in the LP when-The format of our tables is as follows: The first columnever they are added and elimination does not happen toogives the problem name and columns 2–4 and 5–7 giveoften. The exception of test series E is e18 . To the bestthe number of nodes, edges, and terminals of the originalof our knowledge, nobody solved this problem up to nowproblem and the reduced problem, respectively. Compar-to optimality. We are able to solve it within half a daying these two sets of columns reflects the success of ourof CPU time, where Algorithm 3.2 was replaced by apreprocessing algorithm. The next three columns givecomplete reduction test. e18 and d19 are the only exam-statistics about the branch-and-cut algorithm. Nod con-ples of Beasley’s test set where branching was necessarytains the number of branch-and-bound nodes (1 meanswith the default parameter setting.‡that no branching was necessary) , Iter gives the number

Figure 7 depicts a diagram of the run for e16 . Theof cutting plane iterations, and Cuts gives the number ofx-axis shows the number of cutting plane iterations (i.e.,violated cuts added to the LP. The following three col-the number of LPs that have been solved). There wereumns provide information of the root LP, which is the267 iterations for example e16 (see Table II) . The curvesfinal linear program if no branching was necessary and,in the diagram illustrate trends of certain numbers in theotherwise, the last linear program before branching. Fraccourse of the algorithm. The top curve Integer gives thedenotes the number of fractional variables in the root LP,number of variables that are integer in the actual LP solu-and Rows and NZ , the number of rows and nonzeros.tion. In the first LP, all variables are integer, since weThen, time statistics follow: Pre stands for presolve time;start just with the trivial inequalities (see above). Thus,Heu , for the heuristic time; and LP , for the time spentthe straight line indicates that almost all variables areto solve the LPs; the separation time is shown in columninteger during the whole run of the algorithm (we willSep , and, finally, Tot gives the whole running time tosee different patterns for other problems in a moment) .solve the problem. The times are in CPU seconds. TheThe next two curves show the upper and lower bounds.time limit for all runs (with the exceptions of e18,The horizontal line for the upper bound means that thediw0234 , and some incidence problems) was 10,000 sec-first call to the heuristic already found an optimal solution.onds. The last three columns show the solution values.The lowest curve gives the number of rows in the LP. ItsHeu(1) is the value of the solution found by the first callsteady increase demonstrates the effect just mentionedto the primal heuristic, that is, when no linear program-that elimination of inequalities from the LP does not fre-ming solution is at hand (x Å 0). Comparing this valuequently happen. This property is common to many testto the lower bound depicted in column LB provides infor-

mation about the quality of the primal heuristic. If the‡ In fact, branching was only necessary to obtain an optimal solution;

the objective function value of the root LP rounded up already yieldsthe optimal solution value.† URL: ftp: / / ftp.zib.de/pub/Packages/mp-testdata/steinlib.


216 KOCH AND MARTIN

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218 KOCH AND MARTIN

Fig. 9. gap3100.Fig. 7. e16.

examples (see also Figs. 8–10). The bottom bars depict Table IV contains a collection of examples describedin Chopra et al. [8] . E. Gorres made these instancesthe number of simplex iterations to solve the LPs; the

higher the bars, the more difficult were the LPs to solve. available to us. We solved all these instances within sec-onds.§ Interesting to note is that almost always the rootWe see that the number of simplex iterations is high if

there is an increase in the lower bound, and the numbers LP is integer (see Column Frac) .The next series, denoted by R , is taken from Soukupare low when there is no progress in the lower bound. We

have observed this behavior on many Beasley instances. and Chow [37] (see Table V). We solve all of them inabout 1 minute. Worthwhile to note are that in 24 of 38Table III contains some instances made us available

by Margot [33] and some problems on complete graphs. examples the first call to the heuristic already found theoptimal solution and that the LP time dominates all otherbr was introduced in Ferreira [16], whereas berlin and

gr are taken from the TSP library, where some nodes are times. The latter fact seems to be typical for grid exam-ples, which the test set R consists of entirely. This phe-defined as terminals. It turns out that the winning proce-

dure for complete instances is presolve. Algorithm 3.2 nomenon will become clearer in some of the next tests.Tables VI and VII show results for the so-called inci-reduces up to 98% of the edges (variables) and provides

the bases for solving even the big gr example with over dence problems obtained from C. Duin. These problems,described in Duin [12] and Duin and Voß [15], are ran-200,000 variables within 6 minutes.

The diagram of example gr in Figure 8 shows some domly generated and have the following sizes: There arefour sizes of the node set £ :Å ÉVÉ Å 80, 160, 320, anddifferent sign patterns from the one of e16 . We observe

that the number of fractionals (see the curve reflecting 640; for each of them, 20 variants are generated combin-the number of Integers) is low at the beginning, increases ing four sizes of the terminal set ÉTÉ Å log £,

√£ , 2

√£ ,

continuously until the middle of the run, and decreases and £ /4 with five different densities ÉEÉ Å (3£) /2, £ lnagain toward the end. This u-shape behavior is typical £, [£(£ 0 1)]/2, 2£, and [£(£ 0 1)]/10, all values arefor complete instances. We also see that the difficulty of rounded down to the next integer. Every variant wasthe LPs is correlated to the number of fractional variables, drawn five times. The problem names have the patternwhich is also true for the examples depicted in Figures 9 £.tei , where £ Å 80, 160, 320, and 640 gives the numberand 10. of nodes of the problem, t Å 0, 1, 2, 3 indicates which

of the four alternatives (in the above sequence) of thesizes for the terminal sets have been chosen, e Å 0, 1, 2,3, 4 stands for the five densities, and i Å 1, 2, 3, 4, 5distinguishes the five instances drawn for each variant.To give an example, problem 160.141 is the first out offive instances with 160 nodes, [£(£ 0 1)]/10 Å (160r159)/10 Å 2544 edges, and

√£ Å

√160 Å 12 termi-

nals. For each variant, our algorithm behaves very simi-larly for the five instances; thus, we show only the firstin Tables VI and VII. The computations for these prob-

§ The optimal values sometimes differ from the one described inChopra et al. [8] , because they did not add the values of variables fixedby presolve.Fig. 8. gr.



Fig. 10. es40o.

lems were performed on a Sun Enterprise 3000 at a laterdate, using CPLEX Version 5.0 (instead of Version 4.0.9as for the other computations) .

The incidence problems show completely different so-lution characteristics. One main difference is that our pre-solve algorithm (neither the default nor complete test)could not find any significant reductions (note that theproblems were generated with the intention to have diffi-cult problems for presolve, see Duin [12] and Duin andVoß [15]) . On the other hand, the examples do not showthe same sign of degeneracy and we could fix many vari-ables in the course of the algorithm by the reduced costcriterium (see Section 4.5) . To tackle the incident prob-lems, it turned out to be a good idea to restart our branch-and-cut algorithm from scratch after a certain percentageof the variables could be fixed. Column R in Tables VIand VII shows the number of restarts performed and Col-umns 5–7 give, instead of the sizes of the presolvedproblems (which are almost always identical to the origi-nal sizes) , the sizes of the problems after the last restart.We see that with this idea of iterative restarts sometimesa significant amount of variables can be fixed, especiallyif the number of terminals is small (for instance, thenumber of edges of 640.021 can be reduced by 97%).We are able to solve all problems on 80 and 160 nodes.However, we have difficulties to solve some of the largerinstances. There are problems like 320.311 or 640.141that we even cannot solve within 1 day. Table VII presentsthe results, where we used a unique time limit of 10,000seconds for the difficult instances. Within this time limit,we can give a solution guarantee of at least 4.2% for allincidence problems.

What one would like to have at this point is a compari-son with other codes. However, this is very difficult. Peo-ple have different machines with different storage spaces,use different packages for the solution of subproblemslike linear programs, and so on. We refrain from givinga comparison here. The interested reader may refer toBeasley [5] , Chopra et al. [8] , Duin [12], or Lucena[31], who developed comparable codes for the Steinertree problem in graphs. T

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Tab

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cont

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Tables VIII–X give computational results on real-world VLSI instances. One of the challenging problemsin the design of electronic circuits is the routing problem,which is, roughly speaking, the task to connect terminalsets via wires on a predefined area. Depending on theunderlying technology and the design rules, subproblemsarise that can be formulated as the problem of packingSteiner trees in certain graphs (see Lengauer [30] for anexcellent treatment of this subject) . The problems that weare going to consider result from seven different circuitsdescribed in Junger et al. [27]. The underlying graphsare grid graphs that contain holes. The holes result fromso-called cells that block certain areas of the grid. Thesets of terminals are located on the border of these holes.For each of the seven circuits and for each terminal setTi (where index i runs from 1 to the number of terminalsets of the circuit) , we constructed an instance of theSteiner tree problem. For the graph G , we have chosen theunderlying grid graph restricted to the minimal enclosingrectangle of the terminal set. The distance of two neigh-bored grid points in horizontal and vertical directions dif-fer for these circuits. This results in different edge costsfor horizontal and vertical edges in G .

In the library SteinLib, we put all instances with termi-nal sets whose cardinality is at least 10 (in total 475).The examples are distinguished by the name of the circuitfollowed by the index of the terminal set. For example,msm1234 means that the instance is defined by terminalset 1234 of circuit msm . As test problems for our algo-rithm, we chose for each circuit all instances whose twoleading nonzeros of the index of the terminal set differfrom the two leading nonzeros of all other indices. Ifthere are more than one index with the same two leadingnonzeros, we chose the instance with the smallest index(for instance, among examples msm3727, msm3731,msm3761 , and msm3786 , we chose msm3727) . In addi-tion, we added an instance with the smallest and largestnumber of terminals for each circuit. This way we ob-tained 116 different VLSI test instances.

The success of our branch-and-cut algorithm is shownin Tables VIII–X. We solve 83 out of the 116 instances tooptimality within 10,000 seconds and provide a solutionguarantee [(upper bound 0 lower bound)/ lower bound]of less than 10% for 85% of the examples. The biggestwith respect to number of terminals that we solve withinthe time limit are alue5067 and alue6735 with 68 termi-nals each. The biggest in size of the number of edges ismsm3727 with over 8000 edges. However, there are alsosmaller instances, for example, diw0795 with 10 termi-nals or msm2601 with less than 5000 variables after pre-solve, that we do not solve within the time limit. Allruns were performed with the default strategy (except fordiw0234 and alut2625) ; in particular, we applied Algo-rithm 3.2 to reduce the problem and did not perform a

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the

com

plet

ere

duct

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was

used

and

noti

me

lim

itw

asgi

ven.


226 KOCH AND MARTIN

TA

BLE

IX.

VLS

Iex

amp

les:

dm

xan

dm

sm

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00.

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30.

82.

534

934

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344

dmxa

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127

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5510

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89.1

1590

Tab

leIX

cont

inue

s



TA

BLE

IX.

Co

ntin

ued

Ori

gina

lP

reso

lved

B&

CR

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Tim

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135

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440.

10.

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52.

66.

740

439

9.0

399

msm

2492

4045

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1238

1268

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4836

45,3

213.

263

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754.

710

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6014

28.7

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msm

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3031

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1227

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5312

140

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8511

9037

5932

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2.0

28.3

1618

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5.6

1886

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89.1

1290

msm

2601

2961

5100

1627

1148

2916

128

498

3815

9037

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1.8

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1.3

10,0

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800.

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532

8.8

37.3

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734

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111

112

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2597

4935

55,7

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291

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501.

010

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4232

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6.8

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4.1

93.8

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7.7

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560

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84.6

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939

339

2.2

393

msm

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666

1634

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216

139

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380.

738

1

msm

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1115

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138

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2018

0.0

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1.8

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315

310.

831

1

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1048

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385.

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86.3

679.

110

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5516

84.2

2049

msm

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476

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426

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326

1909

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407.

640

8

msm

4515

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1373

413

0613

193

2175

453

1238

11,2

000.

21.

414

8.4

15.8

166.

464

062

9.4

630


228 KOCH AND MARTIN

TA

BLE

X.

VLS

Iex

amp

les:

gap

,al

ue,

and

alu

t

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lP

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Tim

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sN

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eu(1

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gap1

307

342

552

1728

348

517

134

626

118

517

3401

0.0

0.2

2.5

1.7

4.8

554

548.

154

9

gap1

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541

906

1046

581

510

162

932

067

450

060.

10.

411

.34.

416

.645

745

7.0

457

gap1

500

220

374

1716

629

312

169

284

4723

515

460.

00.

20.

70.

71.

925

425

3.2

254

gap1

810

429

702

1735

460

417

138

600

047

630

030.

10.

32.

22.

14.

949

048

2.0

482

gap1

904

735

1256

2167

311

8321

151

1385

131

1088

7755

0.1

1.2

15.4

8.1

25.4

778

762.

676

3

gap2

007

2039

3548

1718

9433

6917

725

866

4560

2487

24,8

500.

919

5.4

2048

.216

3.6

2410

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2011

03.4

1104

gap2

119

1724

2975

2915

5727

7229

185

4427

023

7922

,744

0.6

6.7

748.

360

.381

7.1

1269

1244

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44

gap2

740

1196

2084

1410

8019

3414

114

935

4749

616

2416

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0.3

4.2

578.

939

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3.1

745

744.

374

5

gap2

800

386

653

1232

857

712

165

824

060

344

630.

10.

49.

62.

913

.238

738

6.0

386

gap2

975

179

293

1015

626

710

140

316

027

716

820.

00.

10.

80.

51.

624

524

5.0

245

gap3

036

346

583

1330

853

513

143

912

288

567

4782

0.1

0.3

12.3

2.1

15.1

469

456.

345

7

gap3

100

921

1558

1179

213

9311

115

823

6357

913

0112

,083

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,043

es40

g11

7222

6440

1164

2254

391

140

9181

024

9734

,988

12.9

12.7

3037

.911

0.4

3175

.846

,551

,607

45,6

39,0

09.0

45,6

39,0

09

es40

h12

6224

4440

1254

2436

391

180

12,8

4516

6326

0640

,227

15.9

17.3

9825

.516

8.1

10,0

29.3

49,9

53,7

6348

,739

,666

.448

,745

,996

es40

i12

3223

8440

1228

2380

401

228

14,6

5718

0225

9742

,316

15.1

22.4

9789

.623

1.5

10,0

61.9

52,8

59,3

6951

,557

,587

.251

,761

,789

es40

j12

5524

3040

1251

2426

401

145

11,0

5417

8826

7843

,248

15.1

15.8

9894

.113

9.6

10,0

66.7

58,3

90,8

6256

,761

,892

.057

,414

,203

es40

k11

9223

0440

1187

2299

401

201

12,9

520

2543

41,9

2413

.519

.486

31.1

182.

888

49.3

47,7

19,9

3846

,734

,214

.046

,734

,214

es40

l12

6124

4240

1256

2437

401

178

9872

026

3237

,136

16.1

18.4

4280

.216

0.7

4477

.245

,088

,751

43,8

43,3

78.0

43,8

43,3

78

es40

m13

8126

8240

1377

2678

401

169

10,9

060

2881

41,2

5518

.118

.754

32.0

197.

456

68.3

52,3

74,5

6051

,884

,545

.051

,884

,545

es40

n13

1325

4640

1309

2542

401

195

12,1

2716

1627

7745

,105

16.6

21.6

9793

.219

6.1

10,0

29.7

49,9

67,2

6848

,924

,948

.149

,448

,257

es40

o13

0725

3440

1300

2527

401

232

14,7

120

2575

40,7

6217

.124

.885

51.0

51.7

8847

.251

,340

,298

50,8

28,0

67.0

50,8

28,0

67



limit given, it usually pays to call all reduction methods to instances. We solve many of these instances and providereasonable solution guarantees (in general, below 15%)reduce the problem as much as possible in size. For in-

stance, we solve example diw0234 with over 10,000 vari- for all examples except for the really big ones with severalhundred terminals and tens of thousands of edges. Onables in about 24,000 seconds. The complete presolve

reduces the problem from 10,086 edges to 7266, whereas rectilinear Steiner tree problems, our code performs wellonly for examples with a small number of terminals. To beAlgorithm 3.2 reduces it just to 9991 edges. However,

over 12,000 seconds are spent in presolve when the com- competitive with state-of-the-art software for rectilinearproblems, our reduction methods have to be adapted toplete reduction test is performed and only 6 seconds when

Algorithm 3.2 is applied. (With the default parameter rectilinear instances and more investigations are neces-sary to avoid degenerated linear programs. All examplessetting, we obtain after 10,000 seconds an upper bound

of 1997 and a lower bound of 1967 providing a solution discussed in this paper are gathered in a newly introducedlibrary called SteinLib that is accessible via anonymousguarantee of 1.5%.) The difficulty of the VLSI problems

seem not only depend on the number of terminals, but ftp or the World Wide Web.also on the shape of the grid graphs, how many holes arethere, and how big these holes are. Figure 9 shows atypical diagram for these problems. The numbers of frac- REFERENCEStional variables continuously increase (see the decrease

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[3] M. Ball, W. Liu, and W. Pulleyblank, Two terminallinear problems. Table XI contains the examples fromSteiner tree polyhedra. Contributions to Operations Re-Beasley with 10 and 20 terminals. They are not verysearch and Economics—The Twentieth Anniversary ofdifficult (up to 4 minutes) , although branching is neces-CORE (B. Cornet and H. Tulkens, eds.) . MIT Press,sary in three cases. However, the situation changes forCambridge, MA (1989) 251–284.

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