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Operations Research Letters 32 (2004) 181–184

OperationsResearchLetters

www.elsevier.com/locate/dsw

Separating multi-oddity constrained shortest circuits over thepolytope of stable multisets

Eddie Chenga, Sven de Vriesb;∗

aDepartment of Mathematics and Statistics, Oakland University, Rochester, MI 48309, USAbZentrum Mathematik, TU M$unchen, D-85747 Garching bei M$unchen, Germany

Received 26 February 2003; received in revised form 23 April 2003; accepted 23 May 2003

Abstract

The maximum stable set problem is NP-hard. Koster and Zymolka introduced as a generalization the stable multisetproblem by allowing vertices multiple times subject to vertex- and edge capacities and introduced cycle inequalities. Wederive an e8cient separation algorithm for them.c© 2003 Elsevier B.V. All rights reserved.

Keywords: Stable set problem; Stable multiset problem; E8cient separation algorithm; Odd valued odd cycle inequality

1. Introduction

Let G = (V; E) be a simple connected graph with|V | = n¿ 2 and |E| = m. A subset of V is called astable set if it does not contain adjacent vertices ofG. Let N be a stable set. The incidence vector of Nis x∈{0; 1}V such that xv = 1 if and only if v∈N .(We use {0; 1}V instead of {0; 1}|V | so that such avector is indexed by the elements of V .) The stable setpolytope of G, denoted by STAB(G), is the convexhull of incidence vectors of stable sets of G. Somewell-known valid inequalities for STAB(G) includethe trivial inequalities (xv¿ 0 for v∈V ), the cycleinequalities (

∑v∈C xv6 k where C is the vertex-set

of a cycle of length 2k+1), and the clique inequalities

∗ Corresponding author. Tel.: +49-89-289-16-876;fax: +49-89-289-16-859.

E-mail addresses: echeng@oakland.edu (E. Cheng),devries@ma.tum.de (S. de Vries).

(∑

v∈K xv6 1 where K induces a clique). A cliqueinequality is called an edge inequality if the clique hasjust two vertices.The separation problem for a class C of valid in-

equalities is: Given x∗ ∈RV , does x∗ violate one ofthe inequalities in C? If the answer is yes, exhibit suchan inequality. This problem is important if one wantsto use the inequalities in a branch-and-cut approachto optimize a linear function over STAB(G). (see,for example, Barahona et al. [1] and Nemhauser andSigismondi [8]). Furthermore, if a separation prob-lem is solvable in polytime, then the correspondingoptimization problem can be solved in polytime, see[4].We use N0 to denote N ∪ {0}. A stable multi-

set for a graph G = (V; E) with ∈NV0 ; �∈NE0 isa vector x∈NV0 with xv6 v (vertex inequality) forall v∈V and xv + xw6 �vw (edge inequality) for allvw∈E. Notice that a stable set is a stable multi-set when all ; � equal one. The maximum stable

0167-6377/$ - see front matter c© 2003 Elsevier B.V. All rights reserved.doi:10.1016/j.orl.2003.05.001

182 E. Cheng, S. de Vries / Operations Research Letters 32 (2004) 181–184

multiset problem is then, given weights inNV0 , to Hnda stable multiset with the largest weight. The poly-tope MSTAB(G; ; �) is the convex hull of stable mul-tisets, while EMSTAB(G; ; �) := {x∈RV+:∀v∈V :xv6 v and ∀vw∈E : xv+xw6 �vw} and ESTAB :=EMSTAB(G; 1; 1) where 1 denote the all 1’s vector.Koster and Zymolka [6] reported that while for

ESTAB GrJotschel and Pulleyblank [5] found a sep-aration algorithm for the cycle constraints, no sepa-ration algorithm for cycle inequalities over MSTABis known. In this paper we present an algorithmthat can separate over a generalization of theseinequalities.

2. Preliminaries

As [6] points out, the following assumptionson the ’s and �’s can be made without loss ofgenerality: v ¿ 0 for v∈V , �vw ¿ 0 for vw∈E,0¡max{ v; w}6 �vw ¡ v + w for vw∈E, andminw∈N (v) �vw− w=0 for v∈V . Here, N (v) denotesthe set of vertices adjacent to v. They show that underthese assumptions the dimension of MSTAB equals|V |, that the non-negativity inequalities xv¿ 0 forv∈V deHne facets of MSTAB and that the inequal-ities xv6 v for v∈V deHne facets if and only if�vw ¿ v for all w∈N (v).Next, they consider the cycle inequalities

∑v∈C xv

6 ��(C)=2 for a cycle (simple closed path) C ofG where �(C) =

∑vw∈E(C) �vw and E(C) denotes

the edge-set of C. The validity can be established byadding up the edge inequalities for edges in C, divid-ing the result by 2 and rounding down the right-handside. A cycle is called odd–even if |C| is odd and�(C) is even. We deHne odd–odd, even–odd andeven–even analogously. We will use the term ∗–oddif �(C) is odd and no restriction is placed on |C|.The following result regarding facets is given in [6].

Proposition 2.1. The odd–even, even–even and even–odd cycle inequalities for stable multiset polytopesare implied by vertex and edge inequalities.

Finally, they point out that no e8cient separationalgorithm for the cycle inequalities is known.

3. An observation

We note that by Proposition 2.1, it is enough toseparate over the odd–odd cycle inequalities insteadof all cycle inequalities. In the next section we aregoing to show how to reduce the separation problemfor a generalization of these odd–odd inequalities toa multi-oddity constrained shortest circuit problem. Akey observation for this is to note, that the cycle in-equality

∑v∈C xv6 ��(C)=2 remains valid if C is

a circuit, i.e.: a closed walk. The validity follows asbefore by adding up the edge inequalities for edgesin C, dividing the result by 2 and rounding down theright-hand side. Since C is a circuit, the sum overv∈C has to account for multiple occurrences of ver-tices in C. In other words, if C goes through a vertexw twice then xw is added twice in this sum. The con-cept of extending valid inequalities in this manner isnot new, it has been applied to the stable set poly-topes [2,3]. In fact, such generalization often plays acritical role in designing polynomial-time separationalgorithm.

4. Separating odd–odd circuit inequalities by themulti-oddity-constrained-shortest-circuit-problem

Consider the following problems where d¿ 2 is aHxed positive integer:

MULTI-ODDITY-CONSTRAINED-SHORTEST-CIRCUIT(d)Instance: A graph G=(V; E) with nonnegative

edge costs c, edge weights w∈NE×d0 , a thresholdq∈Q+, and a vertex z ∈V .Output: If there exists a circuit C in G through z

with∑

e∈C ce ¡q such that∑

e∈C we; i ≡ 1mod 2 for16 i6d, then the output is ‘Yes’ with the circuitthat certiHes the answer; otherwise ‘No’.andODD–ODD-CIRCUIT-SEPARATIONInstance: x̂∈EMSTAB(G; ; �).Output: If there exists an odd–odd circuit C in G

with∑

v∈C x̂v ¿ ��(C)=2, then the output is ‘Yes’with the odd–odd circuit that certiHes the answer; oth-erwise ‘No’.A simple computation shows that the ∗-odd

circuit inequalities can be rewritten in the form1=26

∑uv∈E(C) (�uv − xu − xv)=2. Therefore, the

E. Cheng, S. de Vries / Operations Research Letters 32 (2004) 181–184 183

ODD–ODD-CIRCUIT-SEPARATION problem reduces toHnding an odd–odd circuit C (if one exists) with12 ¿

∑uv∈E(C) (�uv − x̂u − x̂v)=2.

Theorem 4.1. ODD–ODD-CIRCUIT-SEPARATION can besolved by solving |V | instances of MULTI-ODDITY-CONSTRAINED-SHORTEST-CIRCUIT(2).

Proof. Set q := 12 and for uv∈E set cuv :=

(�uv − x̂u − x̂v)=2, wuv;1 := 1, and wuv;2 := �uv.Clearly, a violated odd–odd circuit inequality existsif and only if for the given c; w; q a multi-oddity re-stricted circuit of length less than 1

2 exists throughsome vertex z of G. Hence, the problem ODD–ODD-CIRCUIT-SEPARATIONS can be solved by solving |V |instances of MULTI-ODDITY-CONSTRAINED-SHORTEST-CIRCUIT(2) (that is for all z ∈V ).

This by itself would not be progress towards solv-ing ODD–ODD-CIRCUIT-SEPARATIONS if MULTI-ODDITY-CONSTRAINED-SHORTEST-CIRCUIT(2) were hard. Thenext theorem shows that MULTI-ODDITY-CONSTRAINED-SHORTEST-CIRCUIT(d) is not hard (since d is Hxed). Itis solvable by a shortest path computation in a graphwith O(|V |) vertices and O(|E|) edges. The followingnotation is needed in the proof of the next result. Letf1; f2 ∈Nd0 . DeHne f1 ⊕ f2 to be f1 + f2 wheref1 and f2 are treated as vectors over Z2 (so theircomponents are reduced modulo 2) and additions arecarried out over Z2.

Theorem 4.2. The problem MULTI-ODDITY-CONSTRA-

INED-SHORTEST-CIRCUIT(d) for an instance (G; c; w; q; z)is equivalent to a shortest path computation in agraph with |V |2d vertices and |E|2d edges.

Proof. Consider a graphH with vertex set V×Zd2 . Forevery edge uv∈E and any y∈Zd2 we add the edges(u; y)(v; y ⊕ wuv) (that is, (u; y) and (v; y ⊕ wuv) inV × Zd2 are the endpoints) to H and give it cost cuv.Now every multi-oddity-constrained circuit C throughz corresponds to a walk in H from (z; 0) to (z; 1). So,to decide whether a multi-oddity-constrained circuitcontaining the vertex z of cost strictly less than q existsis equivalent to the question whether in H there is a(z; 0)-(z; 1)-walk of cost strictly less than q. The latter

question can of course be decided by one shortest pathcomputation in H .

In fact, application of a bidirectional Dijkstra-algorithm is most appropriate in practice. Further-more, since for distances l in H , l((z; 0); (u; y)) =l((z; 1); (u; y ⊕ 1)) for all (u; y)∈V × Zd2 , additionaltime can be saved as it becomes unnecessary to com-pute the distances from (z; 1). Theorems 4.1 and 4.2together show that the separation problem for odd–odd circuit inequalities can be solved in polytime.Since the Fibonacci-heap Dijkstra algorithm has arunning time of O(|V |log|V | + |E|), we obtain thefollowing result.

Corollary 4.3. ODD–ODD-CIRCUIT-SEPARATIONS canbe solved in O(|V |2log|V |+ |E‖V |) time.

5. Directions for future research

Although there are more ways to derive thepolynomial-time separability, we choose the pre-sented way because the involved transformationvia MULTI-ODDITY-CONSTRAINED-SHORTEST-CIRCUIT(d)looked most promising for future separation applica-tions.Recently, Koster and Zymolka [7] also solved the

cycle separation problem independently via a diRerentapproach.

Acknowledgements

We are grateful to the anonymous referee for anumber of helpful comments and suggestions.

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