An Approach for the Multi-Period Network DesignWith Incremental Routing problem

Technical Report: Université de Technologie de Compiegne,Laboratoire HEUDIASYC.

Benoit Lardeux1,2 and Dritan Nace2

[benoit.lardeux] @francetelecom.com, nace@utc.fr1 Laboratoire CORE/MCN France Telecom R&D,

38-40 rue du Général Leclerc,

92794 Issy-les-Moulineaux, France.2 Université de Technologie de Compiègne,

Laboratoire Heudiasyc, UMR CNRS 6599

60205 Compiègne, France.

Abstract

In this paper we study a special case of Multi-Period NetworkDesignproblems (MPND), Multi-Period Network Design with Incremental routing(MPNDI), as applied in the transmission layer of an optical network. Weoptimize simultaneously the network architecture and the link dimensioningunder discrete cost functions in order to route all traffic demands in line withtraffic growth over a discrete time horizon. Because of operational require-ments, routing paths used at a given period to route some traffic demandshave to be preserved over subsequent periods; this is calledincremental rout-ing with respect to incremental flow. To handle the high dimensionality, acompact formulation omitting explicitly-written flow variables is given anda solution approach taking advantage of the polyhedral structure of MPNDIis proposed. Numerous computational results for a range of instances witheight and ten nodes are given. They show the efficiency of somenew validinequalities proposed in the paper and provide a comparisonof the classicalMPND problem and the MPNDI problem in terms of overall costs and CPUtime.Keywords: Multi-period network design, Integer programmin g, Incre-mental flow, Polyhedral study, Extreme rays.

1 Introduction

A telecommunication company has to provide a functioning network for allcustomers at all times. Network planners must choose a suitable topologyand dimension links to be able to handle traffic routing for all services. Thewell-known network design problem is finding the best architecture and set-ting up bandwidths in links in line with traffic forecasts. Anincreasing num-ber of Internet users exchanging an ever-greater variety ofdata types meansthat the volume of traffic will continue to rise. Telecommunication com-panies seek to forecast reliably the growth in traffic demandover time (indiscrete periods) and to design networks to satisfy this demand, while min-imizing investment costs. Telecommunications hardware depreciates overtime while there is strong modularity and a clear scaling effect that has tobe taken into account. All this leads us to employ a general step function forthe cost of hardware that differs between different periodsand different links.

This work particularly concerns the transmission layer in an optical trans-port network. In such networks Wavelength Division Multiplexing systems(WDM) interconnect the Optical-Electrical-Optical (OEO) cross-connects oreven patch panels. WDM systems provide, for example, 32 wavelengths,each of them supporting the bit rate of 2.5, 10 or 40 Gbit/s. The data rate,as well as the cost of a link, depends on the size of cards installed in the twocross-connects extremities of the WDM system. In our study, we also need todeal with a particular technical requirement: the paths chosen and the band-width capacity used for routing the traffic added at each period should beconserved for subsequent periods; we call thisincremental routing. This re-quirement is typically useful in the transmission layer where numerous con-trols are necessary to ensure network reliability while switching operationstake time and have a high cost. Thus in practice, the new traffic demands haveto be routed into existing free bandwidth capacities or intonewly-created ca-pacities. We shall refer to this problem as the Multi-PeriodNetwork DesignProblem with Incremental routing (MPNDI). Notice that we are limited to thesplittable routing case, that is, a traffic demand is allowedto be split acrossmultiple paths. Although this doesn’t reflect exactly the reality in modernall-optical networks, which require unsplittable routing, we believe that thisassumption applied in the network design context still yields an acceptabledegree of approximation.

More precisely, the MPNDI problem may be defined as follows. Given atime horizon discretized in periods (typically the time unit corresponding tocapacity expansion is more or less one year), a set of nodes, ageneral stepcost function and a set of traffic demand vectors (one value per period) asso-ciated to each commodity (acommodityis defined by an origin node and adestination node), themulti-period network design with incremental routingprobleminvolves designing the architecture and dimensioning the links over

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the time horizon (i.e., installing capacities on the links at the beginning ofeach period) so that the network can carry all traffic demandsthroughout alltime periods and satisfy the incremental routing requirements at a minimumcost. Similarly, the MPND problem that we are referring to, corresponds tofinding an optimal design for the multi-period network such that the incre-mental routing condition is not imposed [21]. This implies that some routedtraffic at a given period may be rerouted into other paths in subsequent pe-riods. Thus, only the feasibility of a multicommodity flow according to theprovided demand matrix over the time horizon is required. Hence, the in-stalled capacities must be sufficient to satisfy only the traffic constraints.

Related work: The MPND problem is known to be NP-hard since it in-cludes as special cases a number of well-known NP-hard combinatorial opti-mization problems [23]. In the literature, the classical network design prob-lem is widely studied, (see for instance [2, 5, 17] and the references therein).A review of models and methods is available in [24]. The first discrete costfunction models were proposed in [11] and [28]. The solutionmethods arebased on algorithms of cutting planes defined by polyhedral properties. In[17], the exact method combines a procedure for multiple constraint genera-tion with integer linear programs solved by CPLEX. A variantof network de-sign problem studied in the literature is the multiperiod network design prob-lem. Although this problem has been around since the nineteen-seventies(see [10] and [29]), the network design expansion is limitedin these mod-els because no links or nodes can be added from one period to the next.The topology remains the same from the first to the last period. A first "ex-act" method based on dynamic programming is proposed in [15], but rout-ing paths have to be known. The complexity and the dynamic nature of theproblem have been studied in [23]. In [16], the problem has been expressedwith integer variables, thanks to an approximation of the link costs modeledby the Kleinrock function. Path variables are chosen to express the multi-commodity flow. Approximate solutions, assessed by a lower bound obtainedby Lagrangian relaxation, are yielded for network instances of 30 nodes forsix periods. In [8], the problem is modeled by general step cost functions andan arc-node formulation of the multi-commodity flow. The Lagrangian relax-ations strengthened by valid inequalities improve by between10% and15%on average the values of lower bounds obtained in [7]. The heuristic based onLagrangian relaxation gives good approached solutions fornetwork instancesof 12 nodes,3 periods and40 demands. Worthy of mention is also [6] wherethe authors study the multi-period pricing network design problem. Theypropose a non-linear model for the problem and a solution approach basedon a decomposition scheme relying on efficient heuristics.

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Contribution of this work: Using traditional multi-commodity flowarc-path formulation to model MPNDI (or MPND) leads to largeILP (In-teger Linear Programming) problems reputed hard to solve even for verysmall instances [17]. To bypass this complexity entailed bythe huge numberof routing paths, we have used a compact formulation taking into consider-ation the requirement of incremental routing without increasing the numberof variables. This is done by applying the Farkas’ lemma to the arc-pathformulation and deducing a generalization of the Japanese theorem [19, 26]for the incremental multi-commodity flow. We define, what we call incre-mental metric cone, which is closely related to the polyhedron dominant ofthe incremental multi-commodity flow and provide a polyhedral study of it.This is the first major contribution of this work. We believe that another con-tribution is introducing new valid inequalities for the MPNDI problem andmaking effective use of them in an exact method. In addition,as MPNDI ismore constrained and has in theory a higher bandwidth overhead than clas-sical MPND, the problem of estimating the additional cost arises. We aim toprovide an answer to this question too.

This paper is organized as follows: in Section 2, we present a"capacity"model of MPNDI involving only integer variables and we extend the result ofthe Japanese theorem in the case of incremental multi-commodity flows anddefine the incremental metric cone. In Section 3, we provide astudy on theextreme rays of the incremental metric cone and formulate some new validinequalities for the MPNDI problem. An exact method based ona multipleconstraint generation procedure using the CPLEX solver is detailed in Sec-tion 4. The valid inequalities previously defined are efficiently employed inthis method. Computational results for several cases of eight and ten nodenetworks for two periods are presented in Section 5.

2 Mathematical formulation

2.1 Notation, definitions and preliminaries

In telecommunications the network is not upgraded continuously throughoutthe time period, but hardware is installed at known dates, typically once ayear. We therefore consider that the time periodT is discrete.T denotes theset of discrete time periodst, and by extension it also denotes the last period.The network is modeled by|T | graphsGt = (N,Et), for all t ∈ T such thatfor each periodt, Gt is composed of the set of nodesN and the set of poten-tiel links Et with capacity equal to the sum of added capacities until period t.ObviouslyEt ⊆ Eτ , τ > t. We usually model the optical transport networksby undirected graphs since the transmission systems are bi-directional in thistelecommunication network layer. Data transmission is modeled by a set ofmulticommodity flows. For more information on multicommodity flows, see

4

vet3evt2

te(x )et

t3

t2e

e

xte

Figure 1: Instance of cost function

Ahuja et al [1]. We consider that forecasted supplementary traffic demandsto route are known for each period. LetKt, t ∈ T be the set of these de-mands at periodt and letdt

k be the forecasted supplementary demand valuefor commodityk ∈ Kt at periodt.

In this paper, we use a general model of step increasing cost functionfor all added capacity at each period (see Figure 1). This cost model canbe applied for several network configurations and especially for an opticaltransport network. The added capacity models pairs of cardsinstalled in twocross-connect nodes interconnected by a WDM system. Letpt(e) be themaximum number of capacity choices to be added on the linke at periodt.

Then, the set of available capacity values is given byV te = {vt0

e , vt1e , ..., v

tpt(e)e },

for all e ∈ Et, t ∈ T , so that0 = vt0e < vt1

e < ... < vtpt(e)e . We associate

to the setV te , the set of costs{γt0

e , γt1e , ..., γ

tpt(e)e }. Each step of the cost

function is modeled by a decision variable. Let(xte)e∈Et

for any t ∈ T de-note the capacity added in linke at periodt andxtα

e for any0 ≤ α ≤ pt(e),the binary variable linked to the capacity choiceα such thatxtα

e = 1 whenvtα

e ≤ xte, andxtα

e = 0 otherwise. Thus,xte =

∑1≤α≤pt(e)

(vtαe −v

t(α−1)e )xtα

e .

This kind of formulation has been reported to be empiricallymore efficientthan that requiring a single capacity variable equal to one while the others arezeros [17]. Finally,φte(x

te) denotes the cost function for any linke at any

periodt. It can be written as:φte(xte) =

∑1≤α≤pt(e)

(γtαe − γ

t(α−1)e )xtα

e .

In the following we recall some elementary definitions of polyhedral ge-ometry and some fundamental theoretical results very useful for the remain-der of the paper. Given anm × n matrix A and a vectorb ∈ IRm (for somem ≥ 0) such thatPh = {x ∈ IRn : Ax ≤ b}; then,Ph of IRn is called apolyhedron. A vectorx ∈ Ph is called apoint of Ph, and a non-null vectorr is called aray of Ph if Ph 6= ∅ andx + βr ∈ Ph for all x ∈ Ph and

5

all β ∈ IR+. A point of Ph is anextreme pointif and only if it cannot beexpressed as a convex combination of any other pair of distinct points ofPh.A ray r of Ph is anextreme rayif and only if it cannot be expressed as a pos-itive combination of any other pairr1 andr2 of rays ofPh wherer 6= δr1,r 6= δr2 for all δ ∈ IR+.

The Farkas’ lemma states that there exists a vectorx ∈ IRn such thatAx ≤ b, x ≥ 0 if and only if∀y ∈ IRm(y ≥ 0) : yA ≥ 0 ⇒ yb ≥ 0. Further,a convex polyhedral cone can be generated by a finite number ofgenerators(Minkowski’s Theorem, [25]).

Another useful result is the so-called Japanese theorem [19, 26], whichgives an alternative characterization for feasible multicommodity flows. Letfirst recall the definition of theMetn cone:

Metn = {π ∈ IRn(n−1)

2+ |πij ≤ πil + πlj , ∀1 ≤ i < j ≤ n, ∀l ∈

{1, ..., n} \ {i, j}}.For more information about this cone, we can refer to [3, 12] or [13]. TheJapanese theorem can then be restated by:

Theorem 2.1. The pair(c, d) is feasible if and only if(c − d)T π ≥ 0 for allπ ∈ Metn.

wherec andd are respectively the capacity and demand functions definedin a given complete graph [13].

2.2 The arc-path formulation

We can now model the MPNDI problem through an arc-path LP formulation.The variable modeling the flow portion of the demandk routed into pathj atperiodt is given byf t

kj . We denoteQtkj the{0, 1} vector modeling the path

j used for demandk at periodt andPt(k) the set of all possible paths forrouting demandk in graphGt. Clearly, a pathj existing in a graphGt alsoexists in all subsequent graphsGτ , τ > t. MPNDI is then modeled by thefollowing integer linear program.

(MILP ) min∑

t∈T

φt(xt) (1)

s.t.

t∑

τ=1

∑

k∈Kτ

∑

j∈Pτ(k)

Qτkj · f

τkj ≤

t∑

τ=1

xτ , ∀t ∈ T (2)

∑

j∈Pt(k)

f tkj ≥ dt

k,∀t ∈ T (3)

f tkj ∈ IR+,∀j ∈ Pt(k),∀k ∈ Kt,∀t ∈ T (4)

xte ∈ V t

e ,∀e ∈ Et,∀t ∈ T (5)

6

>0T

>011

>012

>01k

>021

>022

>02k

>0T1

>0T2

>0Tk

>02

>01

<<

<

<

<

<

<-1 -1..-1

x1

-1 -1..-1

-1 -1..-1

-1 -1..-1

-1 -1..-1

-1 -1..-1

-1 -1..-1

-1 -1..-1

-1 -1..-1

11 Q

12 Q K

Q11 Q1

2 Q K1Q12Q22 QK

2

Q11 Q 2

1 Q K1 Q 1

2 Q22 Q K

2Q1T Q2

T Q KT

x+x

xt=1

t=T

d 11

d 21

d K1

d 12

d 22

d K2

d 1T

d 2T

d KT

1 2

t

Q1

1

1

1

2

2 T

1

2

T

1

2

T

Figure 2: The constraint matrix description

The overall objective function (1) is to minimize the globalsum of thecosts over periods inT . Constraints (2) are capacity constraints, that is, allbandwidth capacities set up at a particular period must havevalues ensuringthe feasible routing of the traffic from the first to this current period. Con-straints (3) ensure that the traffic demands are totally routed for any period.Finally, constraints (4) and (5) show respectively that theflow variables arenon-negative and that only feasible capacities can be installed. The structureof the constraint matrix is given in Figure 21.

This problem involves a very large number of variables (i.e., paths), andone way to solve it is to use Benders’ decomposition and a column generationalgorithm. In the spirit of Benders’ decomposition one can break down theproblem into a master problem involving only integer capacity variables anda satellite problem intended to check the feasibility of thesolution providedby the master. These programs interact together until an optimal solution isachieved. More precisely, the use of Benders’ decomposition results in hav-ing to solve an alternative sequence ofmaster problems(involving the integervariables) and subproblems. Solving a subproblem may provide aBenders’cut which is introduced into the master problem at the next iteration. The

1Qtk in Figure 2 contains all vectorsQt

k,j , j ∈ Pt(k), k ∈ Kt.

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column generation method is currently used to handle such subproblems. Inrésumé, we start from a small number of variables and we attempt to generatenew ones to decrease the value of the objective function. By duality theoryapplied to multicommodity feasibility problems, this can be performed usingseveral sequences of shortest path computations in a graph whose edges aredefined by the dual variables. Even though the shortest path computationscan be done efficiently using, for instance, Dijkstra’s algorithm [14], this ap-proach has the drawback of managing explicitly the routing paths, while ouraim is only to check the capacity feasibility. To get rid of the huge numberof routing paths, we propose a compact model called "capacity formulation"(see also [5]) since it uses essentially capacity variablesas detailed in thefollowing paragraph.

2.3 The "capacity" formulation

In the network design problem, the capacity formulation is obtained by ap-plying the Farkas’ lemma on an "arc-path" model. Flow variables are pro-jected to the capacity variables which become the only necessary variables tomodel the multicommodity flow feasibility. We thus obtain the feasible mul-ticommodity flow polyhedron. Like when deducing the Japanese theorem,we have applied the duality to the arc-path incremental multicommodity flowformulation of MPNDI, as described in the following.We denote asx the added capacity vector for all links and all periods andd the supplementary traffic demand vector for all demands and all periods.Let (πte ∈ R+)t∈T,e∈Et

and(µtk ∈ R+)t∈T,k∈Kt. Then, from the Farkas’

lemma applied in the constraint matrix ofMILP (Figure 2), we can statethe following implication:

∀π ≥ 0,∀µ ≥ 0, (τ=T∑τ=t

πτ ) · Qtkj − µtk ≥ 0 (∀k,∀t,∀j) ⇒

∀π ≥ 0,∀µ ≥ 0, π · x − µ · d ≥ 0 (A)

To satisfy the right-hand-side of the implication, we can rewrite the left-hand-side like:

∀t,∀k, µtk ≤ min1≤j≤|Pt(k)|

{(τ=T∑τ=t

πτ ) · Qtkj} (B)

We now want to characterize the set of all feasible capacity vectors forthe multicommodity incremental flow. From(A) and(B), we can state thepolyhedron dominant of the incremental multicommodity flowXM∗:

XM∗ = {x ∈ IR

∑t∈T

|Et|

+ |∀π ≥ 0,∑t∈T

∑e∈Et

πte

τ=t∑τ=1

xτe ≥ θT (π)}.

so thatθT (π) is expressed:

θT (π) =t=T∑t=1

∑(ij)∈Kt

l∗tij (

τ=T∑τ=t

πτ ) · dtij

wherel∗tij (

∑τ=Tτ=t πτ ) gives the shortest distance betweeni andj when

8

each linke is valued with∑τ=T

τ=t πτ(e). Then, for allt and for allk if werestrictµtk variables to the shortest paths inGt = (N,Et), we can show asin [18], that constraints induced by these variables dominate constraints(B)and thus exactly describeXM∗.

Without loss of generality we suppose that there exists a demand betweenall pairs of nodes in the graphsGt = (N,Et), for all t. Then, |Kt| =n(n−1)

2 andµ ∈ IR|T |·

n(n−1)2

+ . Furthermore, we also stateπ ∈ IR|T |·

n(n−1)2

+ . Ademand or link which is not available is modeled by a link or demand withnull value. Thus, we have:

µt(ij) = l∗tij (

τ=T∑τ=t

πτ ) ∀1 ≤ i < j ≤ n,∀t ∈ T.

Let restrict∑τ=T

τ=t πτ ∈ Metn, for all t ∈ T , and letXM be the follow-ing polyhedron:

XM = {x ∈ IR

∑t∈T

|Et|

+ |∀π ∈ IMetTn ,∑t∈T

∑e∈Et

πte

τ=t∑τ=1

xτe ≥

∑t∈T

∑(ij)∈Kt

πt(ij)·

(τ=t∑τ=1

dτij)}

with IMetTn the cone defined by the set of vectorsπ = (π1, π2, ..., πT ),

so thatτ=T∑τ=t

πτ ∈ Metn, for all t ∈ T . We call IMetTn the incremental

metric cone.

Proposition 2.2. XM∗ = XM

The proof is available in appendix.The above result allows us to give a more tractable formulation of the MPNDIproblem, withxt

e =∑

1≤α≤pt(e)

(vtαe − v

t(α−1)e )xtα

e :

(MILPM) min∑

t∈T

∑

(ij)∈Et

∑

1≤α≤pt(ij)

(γtα(ij) − γ

t(α−1)(ij) ).xtα

ij (6)

s.t.:

x ∈ XM (7)

xt(α−1)ij ≥ xtα

ij ,∀t ∈ T ,∀(ij) ∈ Et,∀α ∈ {2, ..pt(ij)} (8)

xtαij ∈ {0, 1} ,∀t ∈ T ,∀(ij) ∈ Et,∀α ∈ {1, ..pt(ij)} (9)

The constraint (7) ensures that the capacity vector is in thepolyhedrondominant of all feasible incremental multicommodity flows,and constraints(8) stand for evolution of bandwidth capacity set up according to the costfunctions.

Let have a look at the description ofXM . One can notice thatIMetTncone can be generated by a finite number of extreme rays (Minkowski’s The-orem, [25]). Letλ = (λ1, λ2, ..., λT ) be an extreme ray ofIMetTn and

9

MRay be the set of allIMetTn extreme rays. Then, we can state that thefollowing constraints hold and replace the constraints (7)in the formulationof MILPM.

∑

t∈T

∑

(ij)∈Et

λt(ij) · (τ=t∑

τ=1

∑

1≤α≤pτ (ij)

(vτα(ij) − v

τ(α−1)(ij) ) · xτα

ij ) ≥

∑

t∈T

∑

(ij)∈Kt

λt(ij) · (τ=t∑

τ=1

dτij), ∀λ ∈ MRay (10)

Given the above, it would seem interesting to study the polyhedral structureof the IMetTn cone and its extreme rays. Generally speaking, polyhedralcombinatorics studies combinatorial optimization problems with the help ofpolyhedra (see [25] for instance). This approach associates with a given com-binatorial optimization problem the polyhedron defined as the convex hull ofall its feasible solutions. Unfortunately, it is generallynot possible to de-lineate all the linear constraints necessary to describe the convex hull. Thepolyhedron may either be given by a huge (exponential) number of linearconstraints, or simply unknown. Therefore, starting from an (integer) relax-ation of the problem implied by a subset of constraints, we wish to strengthenit by adding some valid inequalities describing the set of feasible solutions(i.e., constraints that cut off non-feasible solutions). However, when dealingwith optimization problems formulated as integer problems, it would seemsensible to aim at reducing the number of calls for solving the relaxed prob-lem. Thus, generating several "good" valid inequalities, or even better, facetsof the solution polyhedron could be of great help. This is also the main ideabehind the solution approach used to solve the MPNDI problem, as detailedin Section 4. Hence, we first need to study the polyhedron dominant of theincremental multicommodity flows and provide a theoreticaljustification forthe valid inequalities of type (10) used in our approach, Section 5.

3 Polyhedral study

In the previous section, we defined theIMetTn cone which is the main math-ematical structure used to describe the polyhedron dominant of all feasiblesolutions for incremental multicommodity flows, denotedXM . Our goal inthis section is characterizing some special classes ofIMetTn extreme rays.The set of vectors inIMetTn are combinations ofMetn vectors. Some prop-erties ofIMetTn which are analogous toMetn are stated without proof inthe following.

Proposition 3.1. Let λ = (0, ...0, λt, 0, ...0) ∈ IMetTn such thatλt 6= 0. λis an extreme ray ofIMetTn if and only ifλt is an extreme ray ofMetn.

Let IMettn be defined likeIMetTn .

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Proposition 3.2. Takeλ = (λ1, ..., λt) ∈ IMettn, t ∈ T . Then(λ1, ..., λt, 0, ..., 0)is an extreme ray ofIMetTn if and only ifλ is an extreme ray ofIMettn.

Proposition 3.3. Letλ = (λ1, ..., λt, ..., λT ) ∈ IMetTn andt ∈ {1...T − 1}such that(λt+1, ..., λT ) 6= 0. λ is an extreme ray ofIMetTn , if (λ1, ..., λt) /∈IMettn or (λ1, ..., λt) = 0.

Generally speaking, letG = (N,E) be any graph andMet(G) be thecone which denotes the projection of the metric cone on the graphG [19]. IfG is a clique ofn nodes, the metric cone is completely described by triangu-lar inequalities.

It is well-known that the cut set is a subclass of theMetn extreme raysand it seems natural to focus in cuts in a graph. LetS be a subset of nodesN = {1...n} andδ(S) ∈ E denotes the cut defined byS, with (i, j) ∈ δ(S)if and only if |S∩{i, j}| = 1. We shall allow ourselves to use the same nota-tion for the incidence vector:δ(S)ij = 1, if |S∩{i, j}| = 1, elseδ(S)ij = 0,∀1 ≤ i < j ≤ n. The cone generated by the cut set is termed the cut cone(Cutn): Cutn = {

∑S⊆N

µSδ(S)|µS ∈ R+} for all S ⊆ N . The cut cone

is the convex hull of the2n−1 − 1 non-null cuts anddim(Cutn) = n(n−1)2 .

The only extreme rays of this cone are the cut vectors. We denote asCut(G)the cut cone related to a graphG.

Several theoretical works and experiments show that the cutcone is agood approximation of the metric cone. In [27], the Seymour theorem statesthat the cut cone and the metric cone are identical if the graph summingthe demand graph and the support graph has no sub-graph contractible to aclique of order 5. See also the complete enumeration of facets of cut and met-ric polyhedra forn ≤ 8 in [7] or the Laurent-Poljack Conjecture [22], [13].In any case, the set ofCutn extreme rays is a subclass ofMetn extreme rays.

In order to take advantage of the structure of the cut cone in the case of themulti-period network design problem with incremental routing, we define:

ICutTn = {λ ∈ IR|T |n(n−1)/2+ |∀t ∈ T,

τ=T∑τ=t

(λτ ) ∈ Cutn}

The Seymour theorem can be extended forICutT (GT ) andIMetT (GT ).Then we deduce the following result, which is elementary andstated withoutproof.

Proposition 3.4. If the graph sum of the demand graph and the supportgraphGT = (N,ET ) has no sub-graph contractible to a clique of order 5,thenICutT (GT ) = IMetT (GT ).

Since extreme rays ofCutn are a sub-class ofMetn extreme rays, we canstate that extreme rays ofICutT (GT ) are extreme rays of theIMetT (GT )cone. We now wish to obtain extreme rays in the multi-period cut cone with

11

incremental routingICutT (GT ). We look for multi-cuts involving consec-utive periods in order to produce new valid multi-partitioninequalities avail-able for the polyhedron dominant of the incremental multicommodity flow(XM). The remainder of this section is devoted to the case of a two-periodnetwork design problem with incremental routing. We denoteasICut2(G2)the cone related to this problem.

ICut2(G2) = {λ ∈ IR2n(n−1)/2+ |(λ1+λ2) ∈ Cut(G2), λ2 ∈ Cut(G2)}

First of all, we seek to extract extreme raysλ = (λ1, λ2) such thatλ1 is anon-null vector andλ2 defines a cut. Thusλ2 is an extreme ray inMetn andfrom Proposition 3.3 we deduce that necessarilyλ1 /∈ Metn. Then,λ2 can

be writtenµ2 · δ(S), with µ2 ≥ 0 andδ(S) ∈ {0, 1}n(n−1)

2 the cut splittingtwo subsets of nodesS ⊆ N andS̄ = N\S. Let CutS (resp.CutS̄) be thecut cone obtained into the subgraph induced by the node subset S (resp.S̄).

Let λδ1 denote theλ1 projection on the space formed by all pairs of nodes

i andj such that(ij) ∈ δ(S), andλδ̄1 its complementary.

Proposition 3.5. If the graph sum of the demand graph and the supportgraph ofG2 = (N,E2) has no sub-graph contractible to a clique of order 5and(λ1, λ2) ∈ ICut2(G2) with λ1 andλ2 non null vectors andλ2 defininga cutδ(S), thenλδ̄

1 ∈ (CutS ∪ CutS̄).

The proof is provided in Appendix.The above result gives a nice indication where to search for extreme rays ofICut2(G2). It would appear sensible to focus on rays(λ1, λ2) such thatλ2

defines a cut whileλ1 defines cuts respectively in the subgraphs derived fromS andS̄.

Theorem 3.6. Take(λ1, λ2) such that following conditions are met:a) λ2 6= 0 andλ2 defines such a cut inG2 thatλ2 = µ2 · δ(S),b) λ1 6= 0, λS

1 and λS̄1 define cuts in the graphs respectively induced byS

andS̄ with λS1 = µ′

1 · δ(S′) andλS̄

1 = µ1” · δ(S”), µ′1, µ

′′1 > 0.

(λ1, λ2) is an extreme ray ofICut2(G2), if and only ifµ′1 = µ1” = 2µ2 and

λ1(ij) = 0 for all (i, j) ∈ δ(S).

The proof is provided in Appendix.We can deduce analogously to above the following result:

Corollary 3.7. Take(λ1, λ2) such that the following conditions are met:a) λ2 6= 0 andλ2 defines such a cut inG2 thatλ2 = µ2 · δ(S),b) λ1 6= 0, λS

1 defines such a cut in the graph induced byS that λS1 =

µ1 · δ(S′),

c) for all (i, j) ∈ S̄2, we haveλ1(ij) = 0.(λ1, λ2) is an extreme ray ofICut2(G2), if and only if µ1 = 2µ2 andλ1(ij) = 0 for all (i, j) ∈ δ(S).

12

4 Solution approach

Let first remark that MPNDI is rather more difficult than the standard MPNDwith step increasing cost function. The main difficulty, in addition to han-dling integer variables and using step cost functions, is the huge number ofconstraints, some of them closely reflecting the incremental nature of routing.In this section we discuss a general solution approach basedon a Benders-like constraint generation procedure [4] for solving MPNDI. This is an it-erative algorithm comprising two main steps like in [17]. Typically, duringan iteration we first solve the relaxed MILPM problem with a subset of con-straints(10). At this stage, if the solution is feasible, we have reached theoptimal solution and the algorithm stops. Otherwise, we obtain a violatedconstraint, which will be added into the current relaxedMILPM . We con-tinue like this until we obtain a solution inXM . In practice adding one validinequality at a time in the master program is not efficient. Onthe basis of thetheoretical results presented in the previous section, we have studied othervalid inequalities of type(10). Their integration in the master program al-lows the model to be strengthened and significantly reduces the computationtime.

Let MRayr be the subset of theIMetTn extreme rays which generatesthe subset of inequalities definingXM added to the relaxed master programMILPMr at iterationr. The master program at iterationr is:

(MILPMr) min∑

t∈T

∑

(ij)∈Et

∑

1≤α≤pt(ij)

(γtα(ij) − γ

t(α−1)(ij) ).xtα

ij (11)

s.t.:

∑

t∈T

∑

(ij)∈Et

λt(ij) · (τ=t∑

τ=1

∑

1≤α≤pτ (ij)

(vτα(ij) − v

τ(α−1)(ij) ) · xτα

ij ) ≥

∑

t∈T

∑

(ij)∈Kt

λt(ij) · (τ=t∑

τ=1

dτij),∀λ ∈ MRayr (12)

xt(α−1)ij ≥ xtα

ij ,∀t ∈ T ,∀(ij) ∈ Et,∀α ∈ {2, ..pt(ij)} (13)

xtαij ∈ (0, 1),∀t ∈ T ,∀(ij) ∈ Et,∀α ∈ {1, ..pt(ij)} (14)

In the following, we detail the constraint generation procedure employedin our approach. Depending on how the generated violated constraints arecombined and added in the current master program, we distinguish severalstrategies. Generally speaking there are three types of violated constraints.The first type concerns the feasibility constraints, that is, we test whether thesolution provided by the master program is feasible. The status of this test(feasible or not) implies the stop condition of the exact method. The metric

13

inequalities as shown in Proposition 3.1 are also of great interest. In partic-ular, |T | linear programs are computed in order to yield the most violatedmetric inequality at each period. We also look for violated cut and multi-cutinequalities (Theorem 3.6 and Corollary 3.7). Some heuristics, essentially in-spired by the Kernighan-Lin algorithm [20], are developed to generate them.They search for violated bipartition inequalities at each period, as well asviolated tripartition and quadripartition inequalities for two consecutive pe-riods. We have also initially introduced in the master someelementary cuts.They are a special case of cut inequalities [5] and require that the sum of thecapacities of the adjacent links at a particular node is not less than the sum oftraffic demands with one of terminals at this node.

4.1 Feasibility constraints

Testing the feasibility of the solution provided by the master program(x̄r) isdone using the following linear program(MSP r):

zr = min∑

t∈T

∑

(ij)∈Et

λt(ij) · (τ=t∑

τ=1

∑

1≤α≤pτ (ij)

(vτα(ij) − v

τ(α−1)(ij) ) · x̄rτα

(ij) ) (15)

s.t.:τ=T∑

τ=t

λτ(ij) ≤τ=T∑

τ=t

λτ(il) +τ=T∑

τ=t

λτ(lj), ∀(i, j, l) ∈ V 3t , i < j, l 6= i, j,∀t ∈ T (16)

∑

t∈T

∑

(ij)∈Kt

λt(ij) · (τ=t∑

τ=1

dτij) = 1 (17)

λ ∈ R|T |n(n−1)

2+ (18)

Let λ̄r denote the optimal solution of(MSP r) at some iterationr. Ob-viously, if zr < 1, x̄r /∈ XM and a new violated constraint is obtained:

∑

t∈T

∑

(ij)∈Et

λ̄rt(ij) · (

τ=t∑

τ=1

∑

1≤α≤pτ (ij)

(vτα(ij) − v

τ(α−1)(ij) ) · xτα

ij ) ≥∑

t∈T

∑

(ij)∈Kt

λ̄rt(ij) · (

τ=t∑

τ=1

dτij) (19)

Elsezr ≥ 1 and x̄r ∈ XM is the optimal solution for MPNDI. SolvingMSP r at each iteration and adding the violated inequality into the constraintset of the current master program ensures convergence to theoptimal solu-tion.

An alternative way to produce feasibility constraints is toseek to mini-mize a slightly different objective function:

∑

t∈T

∑

(ij)∈Et

(

τ=t∑

τ=1

∑

1≤α≤pτ (ij)

(vτα(ij) − v

τ(α−1)(ij) ) · xτα

ij ) −∑

t∈T

∑

(ij)∈Kt

λ̄rt(ij) · (

τ=t∑

τ=1

dτij) (20)

14

The normalization constraint (17) is then replaced by∑

t∈T

∑

1≤i<j≤n

λt(ij) = 1 (21)

This second program frequently yields a violated inequality different fromthe one obtained by(MSP r), thus increasing the number of violated con-straints added to the master at each iteration.

4.2 Metric inequalities

According to Proposition 3.1, the extension ofMetn extreme rays at eachperiod areIMetTn extreme rays. We must then look for violated metric in-equalities with respect to each periodt ∈ T during iterationr and solution̄xr

provided by the master program, by solving the following linear programs,termed(SP r

t ):

z̃tr = min∑

(ij)∈Et

λt(ij) · (τ=t∑

τ=1

∑

1≤α≤pτ (ij)

(vτα(ij) − v

τ(α−1)(ij) ) · x̄rτα

ij ) (22)

s.t.:

λt(ij) ≤ λt(il) + λt(lj), ∀1 ≤ i < j ≤ n, l 6= i, j (23)

∑

(ij)∈Kt

λt(ij) · (τ=t∑

τ=1

dτij) = 1 (24)

λt ∈ Rn(n−1)

2+ (25)

where the constraints(23) ensure thatλt ∈ Metn for all t ∈ T and theconstraint(24) is a normalization constraint. Let̄λr

t be the optimal solutionof (SP r

t ). If z̃tr < 1, the inequality:

∑

(ij)∈Et

λ̄rt(ij) · (

τ=t∑

τ=1

∑

1≤α≤pτ (ij)

(vτα(ij) − v

τ(α−1)(ij) ) · xτα

ij ) ≥∑

(ij)∈Kt

λ̄rt(ij) · (

τ=t∑

τ=1

dτij) (26)

is added into the current master program.

4.3 Cut and multi-cut inequalities

Metric inequalities, as described above, ensure the existence of feasible mul-ticommodity flow for a given capacity and demand function separately inGt

[19, 26]. Another way to test the multicommodity feasibility is to look forbipartition inequalities, i.e., cuts in the graph which maximize the ratios of

15

the demands with respect to the capacities. The problem of the maximumratio cut is a NP-hard one. A heuristic inspired by a Kernighan-Lin algo-rithm, [20], yields good results. This heuristic, called "MaxRatioCut(i,j)",runs|T | · n(n−1)

2 times; that is, for each pair of nodes(1 ≤ i < j ≤ n) ineach ofGt graphs, we look for a maximum ratio cut. The number of vio-lated bipartition cuts is in practice limited to(n− 1) according to Cheng andHu [9]. Obviously metric and bipartition inequalities are related to graphsassociated to separated periods and are valid for both standard MPND andMNPDI problems [21]. Furthermore, as already mentioned, weintroduce aspecial case of bipartition inequalities (called above elementary cuts) wherethe partition has a single node on one side, and the remainingnodes on theother. In practice they are all quite helpful at the beginning of the processbut, for a faster convergence, we need to strengthen the model with valid in-equalities that involve the incremental routing property.

According to some results given in the previous section (seeTheorem 3.6and Corollary 3.7), the tripartition and quadripartition cuts for two consecu-tive periods are extreme rays ofIMetTn ; thus it could be interesting to lookfor violated multi-partition cuts in consecutive periods.Roughly speaking,a tripartition (respectively quadripartition) cut is defined for two periods asfollows: let δ(S) be a cut into the graphG2 = (N,E2) splitting N into twonode subsetsS andS̄; then the tripartition (respectively quadripartition) cutis completed by finding a "cut"δ′(S′) at one node subset inG1 = (N,E1)(respectively two "cuts"δ′(S′) andδ′′(S′′), one for each node subset).λ val-ues are also fixed according to Theorem 3.6 and Corollary 3.7.Thus, valuesassociated to edges inδ(S) are set to1 while for edges inδ′(S′) andδ′′(S′′)they are set to2; for the remaining edges they are null.

Let us now describe how such constraints are generated. Given a solutionx̄r provided by the master program, the problem of computing a tripartitioninvolves finding a multi-cut that divides the node setN into three disjointsubsetsS1, S2, S3 such that:τ=(t−1)∑

τ=1(

∑∀i∈S1,j /∈S1

x̄rτ(ij)+2·

∑i∈S2,j∈S3

x̄τr(ij))+

∑i∈S1,j /∈S1

x̄rt(ij) ≥ dt(S1|S2S3)

with dt(S1|S2S3) =τ=(t−1)∑

τ=1(

∑i∈S1,j /∈S1

dτij +2 ·

∑i∈S2,j∈S3

dτij)+

∑i∈S1,j /∈S1

dtij

At an iterationr, a tripartition inequality for two consecutive periods{(t − 1), t} is assessed according to the ratioρ′rt .

ρ′rt (S1|S2S3) =dt(S1|S2S3)

τ=(t−1)∑τ=1

(∑

i∈S1,j /∈S1

x̄rτij + 2 ·

∑i∈S2,j∈S3

x̄rτij ) +

∑i∈S1,j /∈S1

x̄rtij

16

The inequality is violated ifρ′rt > 1. The problem which consists incomputing the most violated tripartition inequality is also NP-hard. We use aheuristic inspired by the Kernighan-Lin algorithm called "MaxRatio3Cut(i,j,k)".This heuristic computes tripartition inequality of maximal cost according toρ′rt , for each node triplet so that1 ≤ i < j ≤ n, k /∈ {i, j}, with eachone in a separate subset (S1, S2, S3). In the "MaxRatio3Cut(i,j,k)" method,other nodes (except fori, j andk) can move from one node subset to anothersuch that theρ′rt value increases, in order to find the most violated triparti-tion inequality. Finally, quadripartition inequalities can also be computed inan analogous way.

5 Implementation details and computational re-sults

In this section we present two kinds of numerical tests on several networkinstances. They are intended, first, to show the performanceof our approachand the impact of each type of valid inequalities, and second, to give estima-tions of the additional cost when imposing an incremental routing conditionfor MPND. The solution methods are based on the Benders-likeconstraintgeneration procedures described in Section 4. They are implemented in Javausing CPLEX MIP 9.0 and run in a NEC computer with a Pentium 4 proces-sor and 1 Giga octet of RAM.

Let us first consider some implementation details of these approaches.First, we have introduced at the start some elementary cuts into the relaxedmasters, (see Section 4). Furthermore, achieving optimal solutions of relaxedmaster programs for intermediate iterations is not necessary; good feasiblesolutions are sufficient to generate strong inequalities. Thus CPLEX MIPsolving procedure is stopped when a good feasible solution (1% from opti-mality) is obtained. We carry on like this until no violated inequalities arefound. Then, subsequent masters are solved to optimality until the optimalsolution of MPNDI is achieved. We developed and tested threestrategies dif-fering only in the type of valid inequalities consecutivelyadded in the masterprogram:

• The "basic-constraints" strategy is a basic method adding only feasi-bility constraints at each iteration. These constraints are computed bytwo slightly different methods as already detailed in Section 4.1.

• The "2-partitions" strategy involves adding metric and bipartition aswell as feasibility violated inequalities into the currentmaster program.

• In the "2,3 and 4-partitions" strategy, every valid inequality everpresented (feasibility, metric, cut and multi-cut inequalities) is gen-erated and added into the master program. However, the quadripar-tition inequalities are generated only when no tripartition inequalitiesare found.

17

Network instances:The same network instances are used for these twoseries of experiments. Within a given period, the cost of each link exclusivelychanges according to added capacity values and to the distance between thelink’s two extremities. Furthermore, over periods the costof the link de-creases. The cost of devices during the second period is only0.7 times theircost during the first. For all networks, the problem is solvedfor two con-secutive periods and the supply graphs of demands are fully meshed in thetwo periods. We ran our tests in15 network instances, the first nine instancesbeing8-node networks and the last six10-node networks, (see Table 1).

In : Instance numberNbN: Number of nodesNbL : Number of linksAV : Available capacity values

In NbN NbL AV

1 8 28 0..64..1282 8 28 0..64..1283 8 28 0..64..1284 8 28 0..64..1285 8 28 0..64..1286 8 28 0..32..64..96..1287 8 28 0..32..64..96..1288 8 28 0..32..64..96..1289 8 28 0..32..64..96..12810 10 30 0..64..12811 10 30 0..64..12812 10 30 0..64..12813 10 45 0..64..12814 10 45 0..64..12815 10 45 0..64..128

Table 1: Instances description

T: time to reach the optimum in secondsIt : number of solver master programsAC: number of constraints added to the master programs2C: number of generated bipartition inequalities3-4C: number of generated tripartition and quadripartition inequalitiesT2/T3: computing time ratio of the "2-partitions" method over the"2,3 and4-partitions" methodT3/T0: computing time ratio of the CPU time needed to solve the classicalMPND problem, that the same as MPNDI but with incremental routing con-dition relaxed, and the CPU time for the MPNDI problem. The results inTable 2 show that solving MPNDI even for quite small instances is a veryhard task. The number of required iterations remains high even if we tacklethe problem using an efficient constraint generation procedure. In the light

18

basic-constraints 2-partitions 2,3 and 4-partitions T2/ T3/In T It AC T It AC 2C T It AC 2C 3-4C T3 T0

1 24331 314 626 10731 264 659 94 1208 62 389 72 173 8.88 1212 29811 471 940 8767 312 783 106 1975 113 583 86 240 4.44 1323 1604 306 610 426 129 378 83 274 36 333 77 163 1.55 344 1026 187 372 219 61 239 89 175 23 278 72 143 1.25 95 19787 281 560 5824 124 408 100 2237 60 406 82 183 2.60 296 10739 187 372 1656 83 276 85 1174 39 296 68 132 1.41 177 11687 314 626 9958 225 594 103 1728 76 402 81 151 5.76 358 13232 164 326 1212 96 309 85 986 41 361 83 173 1.23 0.729 >24h *** *** 25836 334 858 117 4685 99 545 80 232 5.51 11710 47551 407 812 5278 109 418 130 4960 89 576 114 228 1.06 3811 >24h *** *** 61169 272 777 118 8355 104 588 100 221 7.32 15512 >24h *** *** 69363 475 1215 158 8740 137 821 132 363 7.94 25713 >24h *** *** 50456 222 742 211 26114 96 728 160 329 1.93 25614 >24h *** *** >24h *** *** *** 16901 68 700 168 355 *** 36715 >24h *** *** >24h *** *** *** 49908 76 607 136 280 *** 860

Table 2: Comparisons between constraint generation methods

of the results concerning the "basic-constraints" and "2-partitions" methods,we observe that for the instances that an optimal solution isobtained by bothof them, the "2-partitions" method achieves it on average almost five timesfaster. This confirms the importance of bipartition inequalities in reducingCPU time. We now focus on the "2,3and 4-partitions" method for solvingMPNDI. The addition of multi-cut constraints is also of great benefit be-cause the number of iterations is radically reduced compared to the solutionsobtained by "2-partitions" (by about two thirds) while the number of con-straints does not significantly increase on average. Finally, generation ofmulti-period multi-cuts, described in Section 3, speeds upfurther the solu-tion approach. The "2,3and 4-partitions" method reaches the optimal solu-tions almost four times faster on average than "2-partitions" methods. Noticealso that the incremental routing requirement leads to a very hard problem.As shown in the last column of Table (except instance 8), the same test in-stances for the corresponding classical MPND problem are solved161 timesfaster on average than the MPNDI problem.

The second series of tests is intended to estimate the additional costwhen the incremental routing condition is imposed. Thus, optimal solutionsfor MPNDI are compared with optimal solutions obtained for the standardMPND problem. In our numerical tests (see Table 3) we assess the globalcost difference between the two problems and the efficiency of the Benders-like constraint generation procedures. These results showthat for MPNDIwe need slightly more capacity than for MPND but this difference in termsof device costs is not really significant ( 1.5% on average). Furthermore thisdifference tends to decrease as the number of available capacities increases(see instances 6 to 9).

19

OC: optimal cost of the network design for two periods% Cost: the difference between the optimal solutions obtained forthe MP-NDI and MPND problems expressed as percentage of the solution for theMPND problem.

MPND 2,3 and 4-partitions : MPNDI %In T It AC OC T It AC OC Cost

1 9 9 81 10621 1208 62 389 10756.7 1.282 6 7 63 7516 1975 113 583 7698.6 2.423 7 10 80 8684.7 274 36 333 8797.7 1.34 15 13 100 8288.8 175 23 278 8349.69 0.735 21 9 88 9223.6 2237 60 406 9356.1 1.446 22 15 106 14754.6 1174 39 296 14789.7 0.247 22 17 102 15347.1 1728 76 402 15453 0.698 220 13 94 15262.7 986 41 361 15307 0.299 10 8 79 15146.3 4685 99 545 15276.2 0.8610 97 28 154 11644 4960 89 576 11773.4 1.1111 34 15 110 11723.8 8355 104 588 12019 2.5212 19 9 118 8239.2 8740 137 821 8507.8 3.2613 33 9 121 7868.4 26114 96 728 8002.8 1.7114 42 11 146 7177 16901 68 700 7410 3.2515 42 8 111 9370 49908 76 607 9558.9 2.02

Table 3: Comparisons between solutions with and without incremental routing

6 Conclusion

From an operational point of view, we provide numerous computational andcomparison tests which telecommunication operators are likely to find par-ticularly useful. The main result is that networks designedon basis of incre-mental routing may be used without bringing about a significant impact interms of cost. Furthermore we have studied in this paper a newoptimizationproblem, the multi-period network design problem with incremental routing,and propose an original polyhedral formulation based on a generalization ofthe "Japanese theorem". New valid inequalities defined in two consecutiveperiods are developed to describe the polyhedron of feasible solutions andwe present an efficient solution approach using these inequalities. We haveremarked that the highest computational burden in our simulations is solvingthe masters, with about 85% of global CPU time. The presentedresults con-firm our strategy to reduce the number of master program callsby introducingseveral valid inequalities in order to reduce the global resolution time. How-ever, the method remains computationally heavy and fails toreach optimalsolutions for 12-node network instances constructed similary to above. Fur-ther work is needed to extend the study of the MPNDI feasible polyhedronin order to characterize valid inequalities defined in more than two periods or

20

to find other efficient valid inequalities. The "capacity" formulation has alsoa drawback: in practice we cannot take account of length limitations on therouting paths (which may be necessary to respect delay constraints imposedfor real time traffic), as we could in the arc-path formulation using explicitrouting path variables.

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22

A Proofs

Proposition 2.1.XM∗ = XM .We first show thatXM∗ ⊆ XM , and nextXM ⊆ XM∗.a)XM∗ ⊆ XM . Let bex ∈ XM∗, then for allπ ≥ 0 we have∑t∈T

∑(ij)∈Et

πt(ij)

t∑τ=1

xτij ≥

∑t∈T

∑(ij)∈Kt

l∗tij (

T∑τ=t

πτ ) · dtij . Then, the above

statement is also satisfied for allπ ∈ IMetTn sinceπ ≥ 0. Furthermore, it

is easy to prove that for allπ ∈ IMetTn andt ∈ T ,T∑

τ=tπτ(ij) = l∗t

ij (T∑

τ=tπτ )

holds for anyt. We can then state that for all

π ∈ IMetTn ,∑t∈T

∑(ij)∈Et

πt(ij)

t∑τ=1

xτij ≥

∑t∈T

∑(ij)∈Kt

l∗tij (

T∑τ=t

πτ ) · dtij =

=∑t∈T

∑(ij)∈Kt

T∑τ=t

πτ(ij) · dtij =

∑t∈T

∑(ij)∈Kt

πt(ij) · (τ=t∑τ=1

dτij).

Thus,x ∈ XM .

b)XM ⊆ XM∗. Let us suppose by absurdity that there existsx ∈ XMsuch thatx /∈ XM∗. In other words for allπ ∈ IMetTn ,∑t∈T

∑(ij)∈Et

πt(ij)

t∑τ=1

xτij ≥

∑t∈T

∑(ij)∈Kt

πt(ij) · (τ=t∑τ=1

dτij) and there existsπ ≥

0 such that∑t∈T

∑(ij)∈Et

πt(ij)

t∑τ=1

xτij <

T∑t=1

∑(ij)∈Kt

l∗tij (

T∑τ=t

πτ )dtij .

Let us define

π′T (ij) = l∗T

ij (T∑

τ=T

πτ ), π′(T−1)(ij) = l

∗(T−1)ij (

T∑τ=(T−1)

πτ ) − π′T (ij) and so

on..

π′1(ij) = l∗1ij (

T∑τ=1

πτ ) −T∑

τ=2π′

τ(ij). Therefore, we have for allt ∈ T ,

∀1 ≤ i < j ≤ n,T∑

τ=tπ′

τ(ij) = l∗tij (

T∑τ=t

πτ ) ≤T∑

τ=tπτ(ij). We can easily

show thatπ′ ≥ 0 andπ′ ∈ IMetTn . Let now deduce:∑t∈T

∑(ij)∈Et

π′t(ij)

t∑τ=1

xτij =

∑t∈T

∑(ij)∈Et

(T∑

τ=tπ′

τ(ij))xtij =

=∑t∈T

∑(ij)∈Et

l∗t(ij)(

T∑τ=t

πτ )xtij ≤

∑t∈T

∑(ij)∈Et

(T∑

τ=tπτ(ij))x

tij =

=∑t∈T

∑(ij)∈Et

πt(ij)(t∑

τ=1xτ

ij) <T∑

t=1

∑(ij)∈Kt

l∗t(ij)(

T∑τ=t

πτ(ij))dtij =

=∑t∈T

∑(ij)∈Kt

(τ=T∑τ=t

π′τ(ij))d

tij =

∑t∈T

∑(ij)∈Kt

π′t(ij)(

τ=t∑τ=1

dτij).

Thusx /∈ XM . Contradiction.

23

Proposition 3.5. If the graph sum of the demand graph and the supportgraph ofGT = (N,ET ) has no sub-graph contractible to a clique of order 5and(λ1, λ2) ∈ ICut2(G2) with λ1 andλ2 non null vectors andλ2 defininga cutδ(S), thenλδ̄

1 ∈ (CutS ∪ CutS̄).

As (λ1, λ2) ∈ ICut2(G2), we have(λ1 + λ2) ∈ Cutn. Then forall triplets (i, j, l) ∈ N3, i 6= j, l 6= i, l 6= j, the following statementholds: (λ1 + λ2)ij ≤ (λ1 + λ2)il + (λ1 + λ2)lj . Let λ2 split the graphin two sub-graphs implied by the node setsS andS̄ respectively. From theabove, we have the following inequalities into each subgraph: for all triplets(i, j, l) ∈ S3 (resp. S̄3), i 6= j, l 6= i, l 6= j, λ1(ij) ≤ λ1(il) + λ1(lj). Thus,

λδ̄1 ∈ (MetS ∪ MetS̄). Furthermore, asMet(G2) = Cut(G2) and as the

two graphs implied byS andS̄ are subgraphs ofG2, we haveMetS = CutSandMetS̄ = CutS̄ . Thus,λδ̄

1 ∈ (CutS ∪ CutS̄).

Theorem 3.6.Take(λ1, λ2) such that following conditions are met:a) λ2 6= 0 andλ2 defines such a cut inG2 thatλ2 = µ2 · δ(S),b) λ1 6= 0, λS

1 andλS̄1 define cuts in the graphs implied byS and S̄ respec-

tively withλS1 = µ′

1 · δ(S′) andλS̄

1 = µ1” · δ(S”), µ′1, µ

′′1 > 0.

(λ1, λ2) is an extreme ray ofICut2(G2), if and only ifµ′1 = µ1” = 2µ2 and

λ1ij = 0 for all (i, j) ∈ δ(S).

We first prove the necessary condition for(λ1, λ2) is an extreme ray: as-suming the opposite, we have(λ1, λ2) an extreme ray, butµ′

1 6= 2µ2, orµ′′1 6=

2µ2 or there exists(i, j) ∈ δ(S)|λ1(ij) > 0. We can build two linearly inde-pendent vectors inICut2(G2), λ′, λ′′ with (λ1, λ2) = (λ′

1, λ′2) + (λ′′

1 , λ′′2).

Let us denoteα = min{µ2/4, µ′1/2, µ′′

1/2}. Then, letλ′ = (λ′1, λ

′2) be such

thatλ′1(ij) = α if (i, j) ∈ δ(S′) ∪ δ(S′′) and0 otherwise; andλ′

2 = αδ(S).Let λ′′ = (λ′′

1 , λ′′2) be the difference between(λ1, λ2) and(λ′

1, λ′2). It can

easily be shown that both vectors are inICut2(G2). They are not null andlinearly independent only if one of the following conditions is satisfied:1) µ′

1 6= 2µ2;2) µ′′

1 6= 2µ2;3) ∃(i, j) ∈ δ(S)|λ1(ij) > 0;

Let us now consider the vice-versa. Assuming the opposite: there existtwo linearly independent vectors inICut2(G2), λ′, λ′′ such that(λ1, λ2) =(λ′

1, λ′2) + (λ′′

1 , λ′′2). Sinceλ2 defines a cut inG2 and bothλ′

2 andλ′′2 should

be in Metn, thenλ2 can only be split in the two following ways: 1) intoitself and a null vector; 2) into two linearly dependent vectors weighted byαand(1 − α). On the other hand,λ1 defines two "semi-cuts" inS andS̄ re-spectively and any decomposition such thatλ1 = λ′

1 + λ′′1 necessarily yields

λ′1, λ

′′1 /∈ Metn. We begin with the first case of decomposition ofλ2. If

λ′2 = 0, thanλ′ /∈ ICut2(G2) and thus there is no linear independent de-

24

composition. For the second case,λ2 is the sum of two linearly dependentvectors weighted byα and(1 − α). Then for(λ′

1, λ′2) ∈ ICut2(G2) with

λ′1ij = 0 for all (i, j) ∈ δ(S), links in the same semi-cut must have the

same value. Let us denote themµ′1a andµ′

1b. Furthermore,µ′1a ≤ 2αµ2

andµ′1b ≤ 2αµ2. Similarly, for (λ′′

1 , λ′′2) ∈ ICut2(G2), we obtainµ′′

1a andµ′′

1b, such thatµ′′1a ≤ 2(1 − α)µ2 et µ′′

1b ≤ 2(1 − α)µ2. As the sum of theright-hand sides of each pair of equations gives2µ2 and as we know thatthe sum of left-hand-side should also giveµ′

1 = µ′′1 = 2µ2, we deduce that

all (i.e., four) equations are necessarily equalities. Therefore,(λ′1, λ

′2) and

(λ′′1 , λ′′

2) are linearly dependent. This contradicts our assumption that thesetwo vectors are linearly independent.

25