The Generalized Subgraph Problem:

Complexity, Approximability and Polyhedra

Corinne Feremans, Martine Labbe, Adam Letchford, Juan Jose Salazar

September 10th 2003

Abstract

This paper is concerned with a problem on networks which we call the Generalized Sub-graph Problem (GSP). The GSP is defined on an undirected graph where the vertex set ispartitioned into clusters. The task is to find a subgraph which touches at most one vertexin each cluster so as to maximize the sum of vertex and edge weights. The GSP is a re-laxation of several important problems of a ‘generalized’ type and, interestingly, has strongconnections with various other well-known combinatorial problems, such as the quadraticsemi-assignment, max-flow / min-cut, matching, stable set, uncapacitated facility locationand max-cut problems.

In this paper, we examine the GSP from a theoretical viewpoint. We show that the GSPis strongly NP-hard, but solvable in polynomial time in several special cases. We also giveseveral approximation results. Finally, we examine two 0-1 integer programming formulationsand derive new classes of valid and facet-inducing inequalities that could be useful to developa cutting plane approach for the exact or heuristic resolution of the problem.

1 Introduction

In recent years there have been several works on so-called generalized graph problems, suchas the Generalized Spanning Tree Problem (GSTP) and Generalized Travelling Salesman Prob-lem (GTSP) — see for example Golden, Levy, and Dahl (1981), Fischetti, Salazar, and Toth(1995),(1997), Feremans, Labbe, and Laporte (2002b),(2002a). In all of these problems, one isgiven an undirected graph G = (V,E), where V is a vertex set partitioned into m clusters Vk,k ∈ K = {1, . . . ,m}, and E is the edge set; the task is to find a spanning tree, Hamiltoniancycle, or whatever, which touches exactly (or at most, or at least) one vertex in each clusteroptimizing a given cost function.

More fundamental than a generalized spanning tree or generalized Hamiltonian cycle iswhat Feremans (2001) calls a generalized subgraph (GS). A generalized subgraph is a subgraphG∗ = (V ∗, E∗) of G, not necessarily connected, such that |V ∗ ∩ Vk| ≤ 1 for all k ∈ K. Figure 1shows a graph and a GS. The small numbered circles represent vertices, the ovals represent theclusters and the lines represent the edges. The lines and circles in bold represent the GS.

Consider the following problem: Given a graph G = (V,E), a partition of V into clusters, andweights on the vertices and edges, find a GS of maximum weight. We call this the GeneralizedSubgraph Problem (GSP). Notice that edges of non-positive weight can be deleted, so we assumewithout loss of generality that all edge-weights are positive. However, we do not require vertex-weights to be positive.

In our view the GSP is a ‘fundamental’ problem, in the sense that it is a natural relaxationof the majority of problems of a ‘generalized’ type. Moreover, as we will show, there are also

1

1

2 3

4

5

67

8

Figure 1: Graph with 8 vertices, 4 clusters and 7 edges.

interesting connections between the GSP and several other well-known combinatorial optimiza-tion problems, including the quadratic semi-assignment, max-flow / min-cut, matching, stableset, uncapacitated facility location, max-cut, vertex cover and maximum clique problems. Thisis the motivation for our study of the GSP.

The structure of the paper is as follows. Section 2 reviews results in Feremans (2001)and Section 3 establishes the above-mentioned connections with other known combinatorialproblems. In Section 4 we examine the computational complexity of the GSP. We show that, evenunder very restrictive assumptions, the decision version of the GSP is strongly NP-complete.However, we show polynomial solvability of some special cases of the GSP, including the casewhere the shrunk graph — the graph obtained by shrinking each cluster into a single vertex — isseries-parallel. In Section 5 we examine the issue of approximability. In particular, we show thatthe GSP can be approximated to within a factor of d/2, if the degrees of all vertices in the shrunkgraph are bounded from above by d, and within a factor of 2q, if q is the maximum cardinalityof a cluster. Finally, in Section 6 we examine two 0-1 integer programming formulations of theGSP and derive new classes of valid and facet-inducing inequalities. Conclusions are given inSection 7.

The following notation will be used in the remainder of the paper. For any S1, S2 ⊂ V withS1 ∩ S2 = ∅, E(S1 : S2) denotes the set of edges with one end-vertex in S1 and the other in S2.When S1 contains only a single vertex v, we write E(v : S2) rather than E({v} : S2) for brevity.We also write δ(S) for E(S : V \ S) and δ(v) for δ({v}).

The shrunk graph is denoted by GS = (V S , ES). It is obtained by contracting each clusterinto a single vertex (hence |V S | = m) and by merging all parallel edges into single ones. Thatis, there exists an edge between two vertices of the shrunk graph if and only if there exists atleast one edge between the two corresponding clusters in G.

Other useful definitions from Graph Theory are the following. A cut-vertex (or articulationpoint) in a graph is a vertex whose removal disconnects the graph. A connected graph with nocut-vertices is called biconnected. A maximal biconnected subgraph of a graph is called a block.Any graph can be decomposed into blocks, where each edge appears in exactly one block, andthese blocks form a tree structure. The block decomposition and associated tree structure canbe found in linear time (see Tarjan (1972)).

2

2 Known Results

As pointed out by Feremans (2001), the GSP can be formulated as the following 0-1 integerprogram. Let us consider a variable xe taking the value 1 if and only if edge e is in the GSand similarly for yv and vertex v. Also, the standard convention is used: for any F ⊂ E,x(F ) denotes

∑

e∈F xe and, similarly, for any S ⊆ V , y(S) denotes∑

v∈S yv. Then the GSP isequivalent to:

max∑

e∈E

wexe +∑

v∈V

pvyv (1)

subject to:

y(Vk) ≤ 1 for all k ∈ K, (2)

x(E(v : Vk)) ≤ yv for all k ∈ K and all v ∈ V \ Vk. (3)

xe ∈ {0, 1} for all e ∈ E, (4)

yv ∈ {0, 1} for all v ∈ V. (5)

In this paper we will denote by Pxy(G) the convex hull of feasible solutions to (2)–(5).In Feremans (2001), it is shown that Pxy(G) is full-dimensional and that the inequalities (2)

and (3), along with the trivial inequalities xe ≥ 0 for all e ∈ E, induce facets. Two more complexclasses of inequalities were also introduced. The first are known as odd cycle inequalities (seeFigure 2 for an illustration).

Proposition 1 (Odd Cycle Inequalities, Feremans (2001)). Let c ≥ 3 be an odd integerand let V1, . . . , Vc be clusters. Suppose that each cluster Vk is partitioned into two non-emptysubsets V 1

k and V 2k . Then the odd cycle inequality

c∑

k=1

x(E(V 1k : V 2

(k mod c)+1)) ≤ bc/2c (6)

defines a facet of Pxy(G).

The second are known as odd clique matching inequalities (see Figure 3 for an illustration).

Proposition 2 (Odd Clique Matching Inequalities (OCMI), Feremans (2001)). Letc ≥ 3 be an odd integer and let V1, . . . , Vc be clusters. Suppose that each cluster Vk is partitionedinto c − 1 subsets V l

k for l = 1, . . . , c − 1. If subsets V lk are all non-empty then the odd clique

matching inequalityc−1∑

k=1

c−1∑

l=k

x(E(V lk : V k

l+1)) ≤ bc/2c (7)

defines a facet of Pxy(G).

We would like to point out that it is not necessary for all of the V lk to be non-empty for an

OCMI to induce a facet. Indeed, the odd cycle inequalities already mentioned can be regardedas OCMIs in which some of the V l

k are empty. In fact, the conditions under which OCMIs arefacet-inducing are rather subtle. We explore this issue in Subsection 6.4.

Feremans (2001) also considered the projection of Pxy(G) into x-space, which we shall denoteby Px(G). This is of interest for GSP instances in which the vertex-weights are zero, becausethen the y variables are not necessary. Feremans shows that Px(G) is also full-dimensional andthat the odd cycle and odd clique matching inequalities, together with the trivial inequalitiesxe ≥ 0 for each e ∈ E, induce facets.

3

V 11

V 21

V 22

V 12

V 23

V 13

V 24V 1

4

V 25

V 15

V1 V2

V5 V3V4

Figure 2: Support graph of the odd cycle constraint for c = 5.

V1 V2

V3V5

V4

V 11V 2

1V 31V 4

1 V 12 V 2

2V 32V 4

2

V 13

V 23

V 33

V 43

V 14 V 2

4 V 34V 4

4

V 15

V 25

V 35

V 45

Figure 3: Support graph of the odd clique-matching constraint for c = 5.

3 Related Problems

As already mentioned, the GSP is a natural relaxation for many problems of a ‘generalized’ type.Interestingly, it is also related to several other well-known combinatorial optimization problems.This is the topic of this subsection.

We begin by pointing out a connection between the GSP and the so-called quadratic semi-assignment problem (QSAP), in which a quadratic function of 0-1 variables must be minimizedsubject to the semi-assignment constraints (see Burkard and Cela (1997), Malucelli and Pretolani(1995)). More precisely, this problem can be formulated as

minF (x) :=∑

i

∑

k

fikxik +∑

i

∑

j

∑

k

∑

l

cijklxikxjl

4

subject to

∑

k

xik = 1 for all i

xik ∈ {0, 1} for all i, k.

Then, by considering the following cost transformation

pik = M − fik for all i, k

wijkl = M − cijkl for all i, j, k, l,

where M is a sufficiently large constant, the objective function of QSAP can be replaced by

maxF ′(x) :=∑

i

∑

k

pikxik +∑

i

∑

j

∑

k

∑

l

wijklxikxjl.

This maximization version of QSAP can be seen as a GSP in which the vertices of cluster Vi (foreach i) correspond to variables xik (for all k) and in which each cluster must be visited exactlyonce.

Conversely, a GSP instance can be transformed into a QSAP instance. To see this, notethat, if yu and yv have been set to one, it is always optimal to set xuv to 1. On the otherhand, if either yu or yv is set to zero, then xuv must be zero too. This means that xuv = yuyv

in any optimal solution. Hence, we could formulate the GSP as the problem of maximizingthe quadratic objective function

∑

v∈V pvyv +∑

{u,v}∈E wuvyuyv subject to (2) and (5). In caseclusters have different cardinality, dummy vertices and edges with zero weight can be addedand, again using the standard ‘big M ’ method, we can obtain an equivalent QSAP instance.

This connection with quadratic 0-1 programming proves to be useful in the following way.When |Vk| = 1 for all k, the GSP reduces to an unconstrained quadratic maximization problemin 0-1 variables. In general these problems are NP-hard, but when, as in our case, all ofthe quadratic terms have non-negative coefficients, it can be solved as a max-flow / min-cutproblem. The reverse is also true: any max-flow / min-cut problem can be transformed into aGSP instance with |Vk| = 1 for all k (see Picard and Ratliff (1975) and Rhys (1970)).

On the other hand, when |Vk| = 2 for all k, the GSP becomes strongly NP-hard, see Section4.

Note that the QSAP formulation can be linearized by introducing variables yijkl = xikxjl

and introducing the extra constraints

yijkl ≥ xik + xjl − 1 for all i, j, k, l

yijkl ≤ xik for all i, j, k, l

yijkl ≤ xjl for all i, j, k, l.

The associated integer polytope is equivalent to the so-called partial constraint satisfaction(PCS) polytope studied by Koster, Van Hoesel, and Kolen (1998). According to Koster et al.(1998), many frequency assignment problems described in the literature are particular cases ofthe PCS problem, and therefore also of the GSP.

Koster et al. (1998) propose a class of valid inequalities, the so-called cycle inequalities, forthe PCS polytope. Expressed in terms of the GSP, these inequalities take the following form:

5

V 21

V 11

V 22

V 12

V 23

V 13V 2

4

V 14

V1 V2

V4 V3

Figure 4: Even cycle inequality with c = 4.

1

2

34

56

7

11

11

2

2

3

3

7

44

4

55

55

66

(a) Matching instance (b) GSP instance

Figure 5: Converting a matching instance into a GSP instance.

Proposition 3 (Cycle Inequalities, Koster, Van Hoesel, and Kolen (1998)). Let c ≥ 3be an integer and let V1, . . . , Vc be clusters. Suppose that each cluster Vk is partitioned into twonon-empty subsets V 1

k and V 2k for all k ∈ {1, . . . , c}. Then the cycle inequality

c−1∑

k=1

x(E(V 1k : V 1

k+1)) +c−1∑

k=1

x(E(V 2k : V 2

k+1)) + x(E(V 11 : V 2

c )) + x(E(V 21 : V 1

c )) ≤ c− 1, (8)

is valid for Pxy(G) and Px(G).

See Figure 4 for an illustration of a cycle inequality with c = 4. When c is even, theseinequalities can be shown to induce facets of Pxy(G) and Px(G) using the proof technique ofKoster et al. (1998). However, when c is odd, they are dominated by the odd cycle inequalities(6). Therefore from now on we will call (8) even cycle inequalities and assume that c is even.

Next, we show that the special case of the GSP in which all vertex-weights are zero is inter-mediate in generality between two other well-known problems: the maximum weight matchingproblem of Edmonds (1965c) and the stable set problem (see, e.g., Grotschel, Lovasz, and Schri-jver (1988)). Figure 5 shows that any instance of the matching problem can be converted intoa GSP instance by ‘exploding’ each vertex of degree d into a cluster of d vertices.

To see that the GSP is in turn a special case of the stable set problem, let us say that twoedges are in conflict if they cannot both be used simultaneously in a GSP solution. It is easy

6

{2, 3} {7, 6}

{4, 8}{1, 4}

{2, 7} {3, 7}

{4, 5}

Figure 6: Conflict graph for the GSP instance in Figure 1.

to see that two edges e and f are in conflict if and only if there exists k ∈ K and two distinctvertices u, v ∈ Vk such that e ∈ δ(u) and f ∈ δ(v). So, given a GSP instance defined on a graphG = (V,E), let us define an auxiliary graph GC , called the conflict graph, as follows. Thereis one vertex in V C for each edge in E. Two vertices in V C are connected by an edge in EC

if and only if the two corresponding members of E are in conflict. Then, there is a one-to-onecorrespondence between feasible GSP solutions in G and stable sets in GC . (Figure 6 showsthe conflict graph corresponding to the GSP instance displayed in Figure 1. Vertices in boldrepresent the bold solution in Figure 1.)

Several other well-known problems can also be transformed into the GSP in a natural way.Examples include the uncapacitated facility location problem (UFLP), the max-cut problem, thevertex cover problem (VCP) and the maximum clique problem. The transformations are fairlystraightforward, so we skip the details in the case of the UFLP and max-cut problem. Thetransformations from vertex cover and maximum clique are described and used in Sections 4and 5, respectively.

4 Complexity

In this section we address certain issues related to the complexity of the GSP and some of itsspecial cases. We begin with the bad news that GSP is a strongly NP-hard problem even undervery restrictive assumptions. Indeed, it is not difficult to show, by reduction from the max-cutproblem, that GSP is strongly NP-hard even when |Vk| = 2 for all k and all vertex weights arezero. We will show an even stronger hardness result, by using the vertex cover problem (VCP).The VCP is defined as follows. Given an undirected graph G′ = (V ′, E′), the VCP looks for aminimum-cardinality subset V ∗ ⊆ V ′ such that each edge in E ′ contains at least one vertex ofV ∗.

Theorem 1. The VCP reduces to GSP.

Proof. Let G′ = (V ′, E′) be a VCP instance. Define another undirected graph G = (V,E)by associating a vertex vi in V to each i ∈ V ′, four vertices tie, tje, ui

e and uje in V to each

edge e = {i, j} ∈ E ′, and two edges {tie, vi} and {uie, vi} in E to each vertex i ∈ V ′ and each

edge e = {i, j} ∈ E ′. The clusters are defined by Te = {tie, tje} and Ue = {ui

e, uje} for each

e = {i, j} ∈ E ′, and Vi = {vi} for each i ∈ V ′. The weight of vertices tie and uie are zero,

the weight of vertices vi are −1, and the weight of the edges are +1. Then the problem offinding a minimum-cardinality subset of vertices in V ′ coincides with the problem of finding amaximum-weight GS in G.

Notice that a slightly different reduction yields a GSP instance with zero vertex weights.Instead of taking the singleton clusters Vi for all i ∈ V ′, take clusters of cardinality two by

7

inserting an additional vertex v′i in each Vi. Also add a last cluster with one vertex V0 = {v′0},and the additional edges {v′i, v

′0} with weight equal to one. This gives an instance of the GSP

with zero vertex weights, bipartite, with at most one edge between each pair of clusters.

Corollary 1. The GSP is strongly NP-hard even if:

1. the shrunk graph is bipartite,

2. there is at most one edge between each pair of clusters on opposite sides of the bipartition,

3. all clusters on one side of the bipartition are singletons with unit cost (negative profit),

4. all clusters on the other side of the bipartition contain only two vertices, each with zeroweight,

5. all edges have unit profit.

Corollary 2. The GSP is strongly NP-hard even if:

1. the shrunk graph is bipartite,

2. there is at most one edge between each pair of clusters on opposite sides of the bipartition,

3. all vertices have zero cost,

4. all edges have unit profit,

5. all clusters contain at most two vertices.

Notice that imposing the strongest conditions of the above two corollaries leads to a poly-nomially solvable GSP instance. Indeed, when all clusters on one side of the bipartition containone single vertex with zero weight, the GSP is trivial. So the above two results can be seen asbest possible.

Given these results, it is natural to look for special cases for which the GSP becomes poly-nomially solvable. Our approach for doing this is to specify a number of ‘reduction’ or ‘pre-processing’ operations, together with some simple well-solved GSP classes. If by using thepre-processing operations a GSP instance can be reduced to a small (polynomial) number ofinstances in the well-solved classes, then the original GSP instance is also polynomially solvable.

Here are three simple well-solved cases:

Proposition 4. The GSP is solvable in polynomial time when each vertex v ∈ V has degree atmost one in G.

Proof. When this condition holds, the GSP instance reduces to solving a maximum-weightmatching problem on the shrunk graph where the weight of an edge {u, v} is pu +wuv + pv.

Proposition 5. The GSP is solvable in polynomial time when |Vk| = 1 for all k ∈ K.

Proof. As mentioned in Section 3, this can be reduced to a max-flow / min-cut problem.

Proposition 6. The GSP is solvable in polynomial time if the number of clusters m is boundedby some fixed constant q.

Proof. The number of candidates for V ∗ is clearly O(|V |m), which is polynomial provided thatm is bounded by a constant. Once V ∗ is selected, it is trivial to determine the optimal E∗.

8

Now we present our ‘reduction’ procedures. Our first reduction procedure involves the elim-ination of edges in G.

Procedure 1 (Edge Elimination): For a given pair of clusters Vk, Vl, suppose that there areat least two edges in E(Vk : Vl) which are not adjacent to any edges in E \ E(Vk : Vl). Thenall of these edges, along with their end-vertices, can be removed except the edge {u, v} and thevertices u, v such that pu +wuv + pv is maximized. Of course, even these can be removed whenthis value is non-positive.

A second reduction procedure is capable of eliminating a cluster from certain GSP instances.The effect is to remove vertices of degree two in the shrunk graph. (This procedure correspondsto the series reduction procedure described in Malucelli and Pretolani (1995).)

Procedure 2 (Cluster Elimination): Suppose that there exist three clusters Vi, Vj , Vk suchthat δ(Vi) = E(Vi : Vj ∪ Vk). (That is, the vertices in Vi are connected only to vertices in Vj

and Vk.) Then cluster Vi can be eliminated as follows. For all u ∈ Vj and v ∈ Vk, computethe quantity d(u, v) := maxs∈Vi

max{0, wus + ps + wsv}, where if an edge e 6∈ E then we := 0.Remove Vi and all edges in δ(Vi) from G. For all u ∈ Vj and v ∈ Vk set wuv := wuv + d(u, v). Ifwuv > 0 and {u, v} 6∈ E then upgrade E := E ∪ {{u, v}}.

The third reduction procedure enables us to eliminate a cluster of cardinality one if thevertex-weight is non-negative.

Procedure 3 (Eliminating a Singleton Cluster): Suppose that there exists a cluster Vl

with Vl = {v} and pv ≥ 0. Then, we can eliminate Vl as follows. For each e = {u, v} ∈ E, setpu := pu + we and remove e from E. (We can either leave v as an isolated vertex with weightpv or remove it from V and add pv to the objective.)

The fourth reduction procedure uses the idea of block decomposition mentioned at the endof Section 1:

Procedure 4 (Eliminating a Block): Suppose that the set B ⊂ V S induces a block in theshrunk graph GS and that it is a ‘leaf’ block, i.e., that there is a unique cut-vertex i in the shrunkgraph such that i ∈ B. Let Vk be the cluster corresponding to i and let V (B) denote the set ofvertices in V which map to vertices in B. Then, for all v ∈ Vk, let p(V (B), v) be the optimalprofit for a smaller GSP defined only on the subgraph of G induced by (V (B) \Vk)∪{v}, whereall weights and prizes remain the same apart from pv, which is set to zero. Then the vertices inV (B)\Vk can be removed, along with the associated edges, and the vertex-prize for each v ∈ Vk

updated as pv := pv + p(B, v).

This procedure can be applied iteratively. Indeed, if all blocks have a sufficiently simple struc-ture, we can solve the entire problem in this way. (The tail reduction procedure described inMalucelli and Pretolani (1995) corresponds to a particular case of Procedure 4 when |B| = 2.)

These four procedures, coupled with the three basic classes of well-solved GSP instances,establish the polynomial solvability of a variety of GSP instances.

Theorem 2. The GSP is solvable in polynomial time when for each vertex v ∈ V there existsk(v) ∈ K such that δ(v) ⊆ E(v : Vk(v)).

9

Proof. After Procedure 1 is applied, the GSP has the structure required in Proposition 4, andcan therefore be solved as a matching problem.

Theorem 3. The GSP is solvable in polynomial time when the shrunk graph is a tree.

Proof. Using Procedure 4, this can be reduced to a trivial GSP instance with only two clusters,which is easy by Proposition 6.

More generally,

Theorem 4. The GSP is solvable in polynomial time when the shrunk graph is not contractibleto K4, i.e., when the shrunk graph is series-parallel.

Proof. Application of Procedure 2 to a ‘leaf’ block reduces such a block to only two clusters.Then, applying Procedure 4 eliminates the block entirely. Iterative application of this idea leadsonce again to a trivial GSP instance with only two clusters.

Other still more general results can be obtained by applying all four procedures simultane-ously, in conjunction with Propositions 4 to 6.

5 Approximability

Given that the GSP is NP-hard in general, it is natural to ask whether it can be approximatedto within a constant factor in polynomial time. The following theorem shows that the answeris negative.

Theorem 5. There exists no polynomial time heuristic with any guaranteed performance forthe GSP unless P = NP.

Proof. This is by reduction from the maximum clique problem. The question “Does a graph Gcontain a clique of m vertices?” can be reduced to the question “Does G′ contain a generalizedsubgraph of

(

m2

)

edges?”, where G′ = (V ′, E′) is constructed from G as follows. The vertex setV ′ is partitioned into m clusters, each one being a copy of V . There is an edge in E ′ wheneverthe corresponding edge is in E and the two end-vertices are not in the same cluster. Figure 7illustrates the construction of G′ from a graph G with m = 3. Further each edge of E ′ has aunit weight and each vertex has a profit equal to −((m− 1)/2− 1/m).

The only solutions of this GS instance having a positive value (equal to one) contain exactlym vertices and

(

m2

)

edges, i.e., correspond to cliques in G. Hence, a polynomial heuristic withguaranteed worst case performance for the GSP would also provide a solution to the maximumclique problem.

However, we do have some positive results concerning approximability. In the remainder ofthis section we assume that all vertex weights are non-negative, and that isolated clusters havebeen preprocessed (i.e., the maximum weight vertex in every isolated cluster is selected in asolution).

We begin with the following simple proposition, which relies on a recent result of Berman(2000).

Proposition 7. If all vertex-weights are zero, the GSP can be approximated to within a factorof d, where d is the maximum degree of a vertex in the shrunk graph GS.

10

1 2

3 4

1

234

12

34

1

234

(a) Original graph G with m = 3 (b) GS instance on G′

Figure 7: The maximum clique problem as a decision version of GSP.

1 2

V3

V4

V5

V6

V7

V8V1 V2

Figure 8: GSTP instance such that edge {1, 2} is the center of a 2d− 1 = 7 claw in the conflictgraph.

Proof. A b-claw is a graph with b + 1 vertices, in which the only edges are those connectingone vertex to all other vertices. It can be checked that the conflict graph of a GSP instance is2d-claw-free, i.e., it cannot contain a 2d-claw as a vertex-induced subgraph (see Figure 8). Yet itwas shown by Berman (2000) that the stable set problem in b-claw-free graphs is approximableto within a factor of b/2.

However, we can achieve a better ratio than this, even for more general vertex weights (butstill nonnegative), by exploiting the structure of the problem. First we will show that the GSPcan be approximated to within a factor of (d + 1)/2, i.e., m/2 in the worst-case. Then, by aslightly more complicated argument, we will show that a ratio of d/2 can be achieved.

Recall that GS = (V S , ES) denotes the shrunk graph. Let F denote a set of subgraphs f inGS . For a given e ∈ ES , let F e ⊂ F denote the set of all subgraphs in F which include edge e.Consider the following Linear Program:

r∗(F ) := min∑

f∈F

rf

11

subject to

∑

f∈F e

rf ≥ 1 ∀e ∈ ES ,

rf ≥ 0 ∀f ∈ F.

This LP calls for a fractional covering of the edges of ES by elements of F . We denote by z∗

the optimal value of a GSP and by G∗ = (V ∗, E∗) the corresponding optimal GS. Let zf denotethe optimal value of the modified version of the GSP instance in which all edges have beeneliminated apart from those which maps onto edges in f in the shrunk graph. Further, let

zH(F ) := max{zf : f ∈ F and r∗f > 0},

where r∗f denotes the optimal value of variable rf in the fractional covering problem associatedwith F .

The following lemma provides approximability information about the general heuristic schemeH which amounts to solving (easier) instances of the GSP and considering the worst case.

Lemma 1. r∗(F ) · zH(F ) ≥ z∗.

Proof. Let h(v) be the vertex of the shrunk graph representing the cluster containing vertex v,and let h(e) be the edge of the shrunk graph onto which edge e maps. We have:

z∗ =∑

e∈E∗

we +∑

v∈V ∗

pv

≤∑

e∈E∗

we

(

∑

f :h(e)∈f

r∗f

)

+∑

v∈V ∗

pv

(

∑

f :h(v)∈f

r∗f

)

=∑

f∈F

r∗f

(

∑

e∈E∗:h(e)∈f

we +∑

v∈V ∗:h(v)∈f

pv

)

(9)

≤∑

f∈F

r∗fzf ≤

(

∑

f∈F

r∗f

)

· zH(F ) = r∗(F ) · zH(F ).

The first inequality comes from the fact that (r∗f ) is a fractional cover of the edges and thusalso of the vertices of the shrunk graph Gs. The term between brackets in (9) represents thevalue of a feasible solution to the modified GSP corresponding to f , it is thus bounded by zf .It follows that the weight of the optimal GSP solution is at most r∗(F ) times the weight of thebest solution.

Theorem 6. The GSP can be approximated to within a factor of (d+1)/2 in polynomial time.

Proof. Consider the special subgraph family F consisting of all forests in GS (hence, computingr∗(F ) is a special case of the problem of fractionally covering the elements of a matroid bybases; see Edmonds (1965b)). Using the ellipsoid method and column generation, where columnscorrespond to forests, an optimal basic solution to the fractional covering problem associatedwith F can be found in polynomial time. (Indeed, the pricing problem is in this case to find amaximum weight forest, which is polynomially solvable.) Moreover, in a basic optimal solutionthere are p positive variables r∗f with p ≤ |ES |. Now, for each positive r∗f , we can compute zf

in polynomial time from Theorem 3. Hence zH(F ) can be determined in polynomial time.

12

It now remains to show that r∗(F ) ≤ (d + 1)/2. For this, we use the characterization ofEdmonds (1965b) of fractional matroid covering. Edmonds’ result, translated into our notation,shows that:

r∗(F ) = maxW∈W

{

|ES(W )|

|W | − 1

}

,

where W contains all sets W ⊆ V S which induce connected subgraphs in GS , and ES(W ) is theset of edges in ES with both end-vertices in W . Now, for any W we have:

|ES(W )| ≤min{d, |W | − 1}|W |

2,

and therefore

|ES(W )|

|W | − 1≤

min{d, |W | − 1}|W |

2(|W | − 1)=

min{d, |W | − 1}

2+

1

2min

{

d

|W | − 1, 1

}

≤d

2+

1

2.

Note that the bound of (d + 1)/2 is tight for the polynomial method based on covering byforests described in Theorem 6, even when G is a complete m-partite graph, all vertex-weightsare zero, and all edge profits are 1. The optimum has profit m(m−1)/2, but each of the solutionsfrom a forest has profit only m− 1. So the ratio is m/2, which is (d+ 1)/2.

This approximation result relies on the fact that the GSP is solvable in polynomial time whenGS is a forest. From the previous section, the GSP is also solvable in polynomial time whenGS is series-parallel. This suggests that one might be able to achieve a better approximationratio by seeking a fractional covering of the edges by series-parallel graphs instead of by forests.Unfortunately, it seems unlikely that the resulting LP could be solved efficiently. So far all wehave been able to achieve is the following strengthening:

Proposition 8. The GSP can be approximated to within a factor of d/2 in polynomial time(assuming d > 1).

Proof. The proof goes along the same lines as the previous theorem but instead of using acovering by forests, we use a covering by graphs in which each connected component contains atmost one cycle. The associated independence system is again a matroid, and therefore the LPcan be solved in polynomial time. Moreover, we can solve the necessary modified GSP instancesin polynomial time. Now by Edmonds’ result:

r∗(F ) = maxW∈W

{

|ES(W )|

|W |

}

,

which is easily shown to be bounded above by d/2.

Finally, we give a rather different approximation result which relies on the maximum size ofa cluster rather than properties of the shrunk graph. We first need another technical lemma.

Lemma 2. Let Km be the complete graph with m vertices and FB be the family of bipartitesubgraphs of Km. There exists a solution to the fractional covering problem associated with FB

in which at most 2l−1 variables are positive and equal to 22−l, where l := dlog2 me. Furthermore,these bipartite graphs can be determined in polynomial time.

13

Proof. We first describe the procedure to obtain the bipartite graphs used in the fractionalcover. Assume the vertices of Km are labelled from 1 to m and let M := {1, . . . ,m}. Fori = 1, . . . ,m and j = 1, . . . , l, let b(i, j) denote the value of the j-th bit in a binary encodingof i. The right-most bit comes first. Now, for any subset S ⊆ {1, . . . , l}, denote T (S) := {i ∈M :

∑

j∈S b(i, j) is odd }. We claim that the following bipartite graphs gives us the requiredcovering: for each set S such that |S| is odd, construct a bipartite graph fS in which T (S)constitutes one side of the bipartition. Clearly, there are 2l−1 odd cardinality subsets S andthus 2l−1 such bipartite graphs fS . These bipartite graphs can be determined in polynomialtime since for m = 2l, the number of odd subsets S is m

2 , which is polynomial in m.It remains to prove that each edge of Km belongs to 2l−2 bipartite graphs fS . Let {i1, i2} be

an edge of Km and D := {j : b(i1, j) 6= b(i2, j)}. Then, {i1, i2} belongs to all bipartite graphsfS such that |S ∩ D| is odd and |S \ D| is even. Indeed, assume w.l.o.g. that i1 ∈ T (S). Weknow that

∑

j∈S

b(i1, j) =∑

j∈S∩D

b(i1, j) +∑

j∈S\D

b(i1, j) = |S ∩D| −∑

j∈S∩D

b(i2, j) +∑

j∈S\D

b(i2, j)

is odd and |S ∩D| is odd. Hence,∑

j∈S b(i2, j) must be even, i.e., i2 6∈ T (S) and {i1, i2} is anedge of fS .

Finally, the number of subsets S such that fS contains a given edge is 2|D|−1 ·2l−|D|−1 = 2l−2.Hence the solution

rf =

{

22−l if f is a bipartite graph fS with |S| odd

0 otherwise

is a feasible fractional covering with value 2.

Theorem 7. Let q be the largest cluster cardinality, i.e., q := maxk∈K |Vk|. Then GSP can beapproximated to within a factor of 2q within polynomial time.

Proof. Using Lemmas 1 and 2, we have that 2 · zH(FB) ≥ z∗. The number of bipartite graphswhich need to be considered to determine zH(FB) is 2l−1 ≤ m but each such GSP instance isunlikely to be polynomial. However, a GSP instance whose shrunk graph is bipartite can beapproximated with a factor of q within polynomial time as follows.

Consider the clusters corresponding to one side of the bipartition of V S . We constructinstances in which each of these clusters is reduced to a singleton and such that each vertex ofthese clusters appears in exactly one of these instances. Further, the edge set contains thosehaving both extremities in the instance vertex set. The maximum number of such instances is qand since all vertex weights are nonnegative, Procedure 3 enables us to solve them in polynomialtime. Denote by zH

B the largest optimal value of those instances. Now, using arguments similarto those in the proof of Lemma 1, we can conclude that a bipartite GSP can be approximatedwithin a factor q in polynomial time. Hence q · 2 · zH

B ≥ 2 · zH(FB) ≥ z∗ which completes theproof.

6 Polyhedral Study

In this section we explore further the polytopes Px(G) and Pxy(G). In order to do this it willbe useful to define Px(G) explicitly, rather than implicitly (as the projection into x-space of

14

Figure 9: Support graph of a star inequality defined by (12)

Pxy(G)). This is easily done, using the idea of conflicts between pairs of edges mentioned inSection 3. Indeed, when all vertex-weights are zero, the GSP can be formulated as:

max∑

e∈E

wexe

subject to

xe + xf ≤ 1 for all conflicting pairs e, f ∈ E, (10)

xe ∈ {0, 1} for all e ∈ E. (11)

Then, Px(G) is the convex hull of feasible solutions to (10)–(11). Moreover, as mentioned inSection 3, the polytopes in x-space are intermediate in generality between matching polytopesand stable set polytopes.

6.1 The basic formulation and its projection

The inequalities (2) and (3), together with the non-negativity inequalities on the x variables,define the feasible region of the basic LP relaxation of the GSP. In this section, we examineproperties of this polytope. We begin by considering its projection onto x-space.

Proposition 9. The projection of the polytope

{(x, y) ∈ [0, 1]|E|+|V | : (2), (3) hold }

onto the space of the x variables is defined by the non-negativity inequalities xe ≥ 0 for all e ∈ E,together with the inequalities

∑

v∈Vl

x(E(v : Vk(v))) ≤ 1, (12)

for all l ∈ K and, for each v ∈ Vl, for any k(v) ∈ K \ {l}.

Proof. It suffices to apply Fourier-Motzkin elimination (Martin (1999)) to the GSP formulationgiven by (1)-(5).

We call the inequalities (12) star inequalities, because their support graph (when shrunk) isa star — see Figure 9. A particular and interesting case occurs when k(i) = k for all i ∈ Vj ,when the inequality reduces to x(E(Vj : Vk)) ≤ 1.

Note that the star inequalities are much stronger than the inequalities (10). Indeed, we havethe following proposition.

15

Proposition 10. An inequality of the form x(F ) ≤ 1 induces a facet of Px(G) if and only if itis either a star inequality of the form (12) or a 3-cycle inequality (i.e., an odd cycle inequalityof the form (6) with c = 3).

Proof. Recall that Px(G) is a special kind of stable set polytope. Padberg (1973) showed thatan inequality of the form x(F ) ≤ 1 is facet-inducing for the stable set polytope if and only ifthe vertices in the set F form a maximal clique in the graph. The result follows from simpleenumeration of possible maximal cliques in the conflict graph GC .

Using the ellipsoid method, one can optimize over either of the polytopes mentioned inProposition 9 in polynomial time. This leads us to ask under what conditions these formulationsgive complete descriptions of Pxy(G) and Px(G). In order to answer this question, we will needthe following theorem, which is of independent interest.

Theorem 8. Let G = (V,E) be the graph associated with a GSP instance. If the shrunk graphGS is not biconnected, then a complete description of Pxy(G) is given by a complete descriptionof the polytopes associated with each block.

Proof. By induction, it suffices to prove the following: let G1 = (V 1, E1) be the subgraphof G associated with a leaf block in GS , let Vk ⊂ V 1 be the cluster corresponding to a cutnode in GS , and let G2 = (V 2, E2) be the subgraph of G induced by (V \ (V 1 \ Vk)). Thena complete description of Pxy(G) is given by a complete description of Pxy(G1) and Pxy(G2).The proof relies on Procedure 4 (eliminating a block). Let x1 (respectively x2) be the subvectorcorresponding to the edges of E1 (respectively E2) and y1 (respectively y2) the subvector forvertices in V 1\Vk (respectively V 2\Vk). Subvector y

k corresponds to the vertices of V k. Finallythe associated objective function coefficient subvectors are denoted by w1, w2, p1, p2 and pk.Let (x∗1, x∗2, y∗1, y∗2, y∗k) be an optimal solution to

maxw1x1 + w2x2 + p1y1 + p2y2 + pkyk (13)

(x1, y1, yk) ∈ Pxy(G1) (14)

(x2, y2, yk) ∈ Pxy(G2) (15)

From (14) it follows that the subvector (x∗1, y∗1, y∗k) is a convex combination of, say t, vertices

(xi1, yi1, yik) of Pxy(G1), i.e.:

(x∗1, y∗1, y∗k) =t∑

i=1

λi(xi1, yi1, yik)

where λi ≥ 0 and∑t

i=1 λi = 1. All those vertices have integer coordinates and either yi(Vk) = 1or yi(Vk) = 0. Hence,

∑

{λi : yiv = 1} = y∗v (16)

and∑

{λi : yi(Vk) = 0} = 1− y∗(Vk) (17)

Further, each vertex (xi1, yi1, yik) must be an optimal solution to

maxw1x1 + p1y1 (18)

(x1, y1, yk) ∈ Pxy(G1) (19)

yk = yik (20)

16

because otherwise, replacing (xi1, yi1, yik) by the optimal solution to (18)–(20) in the convexcombination would yield a solution to (13)–(15) with a better objective function value than(x∗1, x∗2, y∗1, y∗2, y∗k). Recall from Procedure 4 in Section 4 that p(V 1, v) denotes the optimalvalue of (18)–(20) when yi

v = 1. Let p(V 1 \ V k) denote the optimal value of (18)–(20) whenyi(Vk) = 0. We now have that

w1x∗1 + p1y∗1 = w1(t∑

i=1

λixi1) + p1(

t∑

i=1

λiyi1) =

∑

v∈Vk

∑

i:yiv=1

λi(w1xi1 + p1yi1) +

∑

i:yi(Vk)=0

λi(w1xi1 + p1yi1) =

∑

v∈Vk

∑

i:yiv=1

λip(V1, v) +

∑

i:yi(Vk)=0

λip(V1 \ Vk) =

∑

v∈Vk

y∗vp(V1, v) + (1−

∑

v∈Vk

y∗v)p(V1 \ Vk),

using (16) and (17).In consequence, problem (13)–(15) can be rewritten as:

maxw2x2 + p2y2 +∑

v∈Vk

(pv + p(V 1, v)− p(V 1 \ Vk))yv + p(V 1 \ Vk)

(x2, y2, yk) ∈ Pxy(G2).

This theorem implies that the shrunk support graph of an inequality inducing a facet ofPxy(G) must be biconnected. It also immediately yields the following:

Corollary 3. The polytope Pxy(G) is completely described by the non-negativity inequalities,constraints (2) and (3) if the shrunk graph is a forest.

Proof. When the shrunk graph is a forest, each block of the shrunk graph is merely an edge.The result is trivially true when the shrunk graph is an edge, so the result follows from Theorem8.

Note that the condition in Corollary 3 is sufficient but, for general graphs G, it is notnecessary. (It is easy to find counter-examples.) On the other hand, for what we call full GSPinstances, we can obtain a necessary and sufficient condition.

Definition 1. A GSP instance (and its associated graph G) is said to be full if the followingholds for each pair Vk, Vl of clusters: either every vertex in Vk is connected to every vertex inVl, or E(Vk : Vl) is empty.

Theorem 9. Let G be full. The inequalities in the IP formulation of the GSP completely describePxy(G) if and only if all cycles of the shrunk graph involve only singleton clusters (equivalently,each block of the shrunk graph is either an edge or is formed entirely by singleton clusters).

17

Proof. When the shrunk graph is an edge, the result is trivially true. When all clusters aresingletons, then the inequalities in the formulation give a complete description because thecorrespondent matrix is totally unimodular (as mentioned in Section 3, the GSP is a min-cut /max-flow problem). Sufficiency then follows from Theorem 8.

On the other hand, if there is a cycle in the shrunk graph which contains at least one non-singleton cluster, then there is a facet-inducing odd ring inequality. (These inequalities will beintroduced in Subsection 6.3).

In the case of Px(G), a result analogous to Theorem 8 does not hold; the instance on the leftof Figure 13 in Subsection 6.3 gives a counter-example since the support graph of this inequalitycontains edges in different blocks. Nevertheless, we have the following two results.

Theorem 10. The polytope Px(G) is completely described by the non-negativity and star in-equalities if the shrunk graph is a forest.

Proof. Follows from Corollary 3 and Proposition 9.

Theorem 11. Let G be full. The star and non-negativity inequalities completely describe Px(G)if and only if there is no cycle in the shrunk graph containing only non-singleton clusters (equiv-alently, the non-singleton clusters induce a forest in the shrunk graph).

Proof. Given a full graph G meeting this condition, perform the following operation to eachsingleton cluster which has degree d > 1 in the shrunk graph. Suppose without loss of generalitythat the cluster, say {v}, is connected in G to clusters V1, . . . , Vd. Replace the cluster {v} by dnew singleton clusters, say {vi} for i = 1, . . . d. Then, replace each edge {u, v}, where u ∈ Vi,with the edge {u, vi}. It can be easily seen that the polytope Px(G) is unchanged by thistransformation (apart from a re-labelling of edges). The transformed graph meets the conditionof Theorem 10, and therefore Px(G) is completely described by the star and non-negativityinequalities.

On the other hand, suppose that there is a cycle of non-singleton clusters in G. If thiscycle has odd cardinality, then there is a facet-inducing odd cycle inequality of the form (6). Ifthis cycle has even cardinality, then there is a facet-inducing even cycle inequality of the form(8).

Notice that the condition for Px(G) in Theorem 11 is weaker than the condition for Pxy(G)in Theorem 9.

A related question is whether the upper bounds obtained by optimizing over the polyhedramentioned in Proposition 9 are of good quality in general. Some experiments on small GSPinstances have led us to believe that, in the case of Px(G), this issue is connected to the quantity(defined in the previous section) r∗(F ), where F denotes the set of all forests in GS :

Conjecture 1. In the case of no vertex weights, the upper bound obtained using the star andnon-negativity inequalities is never greater than r∗(F ) times the optimum, and, provided thatthe number of vertices in each cluster is ‘sufficiently large’, this ratio is achievable.

In the case of general vertex weights, we have examples where the upper bound using (2)and (3) is positive, yet the optimum is zero. Therefore no similar result holds for Pxy(G).

18

6.2 Some lifting theorems

In this subsection, we present some lifting results which enable new valid (and facet-inducing)inequalities to be derived from known ones. When presenting these results we assume forsimplicity that G is a complete m-partite graph. That is, if clusters Vi and Vj are distinct, thenevery vertex in Vi is connected to every vertex in Vj .

In order to motivate the first lifting result, we will need the following lemma, which is alsoof independent interest:

Lemma 3. All facet-inducing inequalities for P xy(G) can be written in the form αx− βy ≤ γ,with α, β, γ ≥ 0, apart from the non-negativity inequalities xe ≥ 0 and the inequalities

∑

i∈Vkyi ≤

1.

Proof. Clearly, we can always assume that the inequality is in ≤ form. Moreover, the origin isfeasible, which implies that γ ≥ 0.

Now let e ∈ E be an arbitrary edge. If we have an integer solution which lies on the facet,and xe = 1, we can obtain another feasible integer solution by setting xe = 0. Therefore αe ≥ 0.On the other hand, if every integer solution on the facet satisfies xe = 0, the facet must beinduced by the non-negativity inequality xe ≥ 0 (given that P xy(G) is full-dimensional).

Similarly, if we have an integer solution which lies on the facet, and there is a cluster k suchthat yi = 0 for all i ∈ Vk, we can obtain other feasible integer solutions by setting yi = 1 foreach i ∈ Vk. Therefore βi ≥ 0 for all i ∈ Vk. On the other hand, if every integer solution on thefacet satisfies

∑

i∈Vkyi = 1, the facet must be induced by the inequality

∑

i∈Vkyi ≤ 1.

Proposition 11 (Trivial Lifting for Pxy(G)). Suppose that an inequality αx− βy ≤ γ, withα, β, γ non-negative, is valid for Pxy(G). Let G′ be a graph obtained from G by adding anothercluster Vk+1, consisting of a single vertex w (together with the |V | extra edges required to makeG′ complete (m+ 1)-partite). Then the inequality α′x− β′y ≤ γ is valid for Pxy(G′), where α′

is defined by α′e := αe (if e ∈ E), α′

e := 0 (otherwise), and β′ is defined by β′v := βv (if v ∈ V )

and β′w := 0. Moreover, the new inequality induces a facet of Pxy(G′) if the original induced a

facet of Pxy(G).

Proof. Validity follows from the fact that a subgraph of a generalized subgraph is itself a gen-eralized subgraph. Now suppose that αx − βy ≤ γ induces a facet of Pxy(G). Then (giventhat Pxy(G) is full-dimensional) there are |E|+ |V | affinely independent generalized subgraphsin G whose incidence vectors (x∗, y∗) lie on the facet. From these it is trivial to construct|E|+ |V | affinely independent generalized subgraphs in G′ whose incidence vectors (x∗, y∗) sat-isfy α′x∗ − β′y∗ = γ, y∗w = 0 and x∗e = 0 for all e 6∈ E. Note also that, for every v ∈ V , thereis at least one GS among these |E| + |V | GSs which contains vertex v. (Otherwise, all pointsof Pxy(G) in the facet would also satisfy x(δ(v)) = 0, which is impossible because Pxy(G) isfull-dimensional.)

To prove the proposition, we need |V | + 1 more such affinely independent generalized sub-graphs in G′. One of these is found by taking any of the previous solutions and setting y∗w = 1.The remaining |V | are constructed as follows: for each v ∈ V , choose one of the |E| + |V | GSswhich contains vertex v, and add vertex w and edge {v, w} to the GS.

Proposition 12 (Trivial Lifting for Px(G)). Suppose that an inequality αx ≤ β, with α andβ non-negative, is valid for Px(G). Let G′ and α′ be defined as in Proposition 11. Then theinequality α′x ≤ β is valid for Px(G′), and it induces a facet of Px(G′) if the original induced afacet of Px(G).

19

Proof. Similar to the proof of Proposition 11.

Proposition 13 (Vertex Cloning for Pxy(G)). Suppose that an inequality αx − βy ≤ γ,with α, β, γ non-negative, is valid for Pxy(G). Let Vk be a specified cluster and let w ∈ Vk bea specified vertex. Let G′ be the graph obtained from G by ‘cloning’ vertex w. That is, a newvertex w′ is added to Vk and |V \Vk| extra edges are added to make G

′ complete m-partite. Thenthe inequality α′x − β′y ≤ γ is valid for Pxy(G′), where α′ is defined by α′

e := αe (if e ∈ E),α′

e := αuw (if e = {u,w′} for some u ∈ V \ Vk), and β′ is defined by β′v := βv (if v ∈ V ) and

β′w′ := βw. Moreover, the new inequality induces a facet of P

xy(G′) if the original induced afacet of Pxy(G).

Proof. Validity follows from the fact that vertices w and w′ cannot both appear in a GS andby symmetry. The preservation of the property of being facet-inducing follows from a similarargument to that given in Proposition 11.

Proposition 14 (Vertex Cloning for Px(G)). Suppose that an inequality αx ≤ β, with α, βnon-negative, is valid for Px(G). Let G′ and α′ be defined as in Proposition 13. Then theinequality α′x ≤ β is valid for Px(G′), and it induces a facet of Px(G′) if the original induced afacet of Px(G).

Proof. Similar to the proof of Proposition 13.

Theorem 12 (Cluster addition for Pxy(G)). Suppose that an inequality αx− βy ≤ γ, withα, β, γ non-negative, is valid (facet-inducing) and not of type xuv ≤ yu for P

xy(G). Let G′ bethe graph obtained from G = (V,E) by adding another cluster Vk+1 = {w, q} together with edgesin E(Vk+1 : V ) to make G′ complete (m + 1)-partite. Let {u, v} be an edge of E such thatαuv ≥ 0 then α′x − β′y ≤ γ′ is valid (facet-inducing) for Pxy(G′) if α′

e = αe, ∀e ∈ E\{{u, v}},α′

uw = α′vw = αuv, α

′qi = 0, ∀i ∈ V , α′

uv = 0 and β′w = αuv, β

′q = 0, β′

i = 0, ∀i ∈ V .

Proof. The validity follows from the fact that if αx− βy + αuvxuv ≤ γ is valid for Pxy(G) thenαx− βy + αuv(xuw + xvw − yw) ≤ γ is valid for Pxy(G′), where (α, β) and (x, y) are associatedwith G = (V , E) for V = V and E = E\{{u, v}}. Indeed, let (x∗, y∗) be the vector associatedwith a feasible GS solution of G′ with x∗uw = x∗vw = 1 (then y∗w = 1), we have αx∗−βy∗+αuv ≤ γsince (x∗, y∗, 1), where the last component denotes xuv, is a feasible GS solution of G.

The proof of facet-inducingness is done in three steps. First, we prove the result whenG′ = (V ′, E′) is such that V ′ = V ∪ {w} and E ′ = E ∪ {{u,w}, {v, w}}. Then we extendthe result to the case where δ(w) is complete by lifting optimally on the remaining variablesxzw, z ∈ V \{u, v}. The third step is obtained for G′ = (V ′, E′) where V ′ = V ∪ {w, q} andE′ = E ∪ δ(w) ∪ δ(q).

For v ∈ V , let h(v) denote the index of the cluster containing v.

Step 1. Let S be a set of dim(Pxy(G)) affinely independent vectors (x, y) of Pxy(G) such thatαx−βy = γ. Let G′ = (V ′, E′) be such that V ′ = V ∪{w} and E ′ = E ∪{{w, u}, {w, v}}.We associate a vector (x′, y′) of Pxy(G′) such that α′x − β′y = γ′ with each vector (x, y)from S. This extension is made as follows: x′f := xf for f ∈ E\{{u, v}}, x′uv := 0,x′uw = x′vw = y′w := xuv and y′t := yt for t ∈ V . In other words, if edge {u, v} is presentin the solution corresponding to (x, y) in G, in the extension, edge {u, v} is erased andreplaced by edges {u,w} and {v, w} in G′. If edge {u, v} is not present in the vector in Gthen nothing is added in the extension in G′ (see Figure 10).

20

u

u

v

vw

(a) If {u, v} 6∈ E.

u

u

v

vw

G

G′ G′

(b) If {u, v} ∈ E.

G

Figure 10: First extension from G to G′.

u

u

v

vw

(a) Vector (xa, ya).

u

u

v

vw

G

G′ G′

(b) Vector (xb, yb).

Gu

u

v

vw

G′

(c) Vector (xc, yc).

G

Figure 11: Second extension from G to G′.

Since adding vertex w and the two edges {u,w} and {v, w} to G implies that three addi-tional variables are introduced, we need therefore three more vectors. In order to do this,we select three special vectors in S.

1. The first vector (xa, ya) is such that yau = 1 and xa

uv = 0. Such a vector exists in Ssince αx−βy ≤ γ is facet-inducing and S cannot be contained in another hyperplane(in this case, the hyperplane would be xuv = yu). This vector is extended to G′ bysetting x′′auw := 1 and y′′aw := 1 (see Figure 11 (a)).

2. The second vector (xb, yb) is such that ybv = 1. It is possible since S cannot be

contained in another hyperplane (in this case, the hyperplane would be xuv = yv).This vector is extended to G′ by setting x′′bvw := 1 and y′′bw := 1 (see Figure 11 (b)).

3. The third vector (xc, yc) is such that xcuv = 1. It is possible since S cannot be

contained in xuv = 0. This vector is extended to G′ by setting x′′cuw = x′′cvw = y′′cw := 1(see Figure 11 (c)).

These dim(Pxy(G′)) vectors are affinely independent, they correspond to feasible GS so-lutions and they satisfy α′x − β′y = γ′. Indeed, let a generic inequality be λx − µy +λuwxuw + λvwxvw − µwyw ≤ ν (where the notations are the same as in the validity part).Using the first dim(Pxy(G)) vectors defined above, we have λ = Cα, µ = Cβ, ν = Cγand λuw + λvw − µw = Cαuv. Using the last three vectors (xa, ya), (xb, yb) and (xc, yc)that have been extended in two different ways to G′ we obtain λuw = µw, λvw = µw andλuv = 0.

Step 2. The second step of the proof is reached by considering the lifting of the facet-inducinginequality for Pxy(G′) where V ′ = V ∪ {w} and E ′ = E ∪ {{u,w}, {v, w}}. This liftingis done optimally with respect to variable xzw for all z ∈ V \{u, v}. We show that theoptimal lifting coefficient for each xzw is zero for z ∈ V \{u, v}.

21

u vG

z

u vG′

z

w

(a) If {u, v} 6∈ E.

u vG

z

u vG′

z

w

(b) If {u, v} ∈ E.

z vG

u

z vG′

u

w

(c) If z ∈ Vh(u).

Figure 12: Third extension from G to G′.

1. If z 6∈ Vh(u), there exists a vector (x, y) in Pxy(G) such that αx − βy = γ withxuz = 1. This vector can be extended to (x′, y′) in Pxy(G′) where V ′ := V ∪{w} andE′ := E′ ∪ {{z, w}} with E ′ initialized to E ∪ {{u,w}}, {v, w}} by setting x′e := xe

for e ∈ E, y′i := yi for i ∈ V and x′zw = x′uw = y′w := 1, x′, y′ := 0 otherwise. In casexuv = 1 then x′vw is set to 1. These extensions satisfy α′x − β′y = γ′ (see Figure 12(a) and (b)).

2. If z ∈ Vh(u), there exists a vector (x, y) in Pxy(G) such that αx − βy = γ withxvz = 1. Extending this vector to (x′, y′) in Pxy(G′) where V ′ := V ∪ {w} andE′ := E′ ∪ {{z, w}} with E ′ initialized to E ∪ {{u,w}}, {v, w}} is done by settingx′e := xe for e ∈ E, y′i := yi for i ∈ V and x′zw = x′vw = y′w := 1, x′, y′ := 0 otherwise(see Figure 12 (c)).

Step 3. The third step of the proof remains to show that α′x − β′y ≤ γ′ is facet-inducing for G′

when V ′ = V ∪ {w, q}, q ∈ Vh(w), E′ = E ∪ δ(w) ∪ δ(q). To prove this, we show that the

lifting coefficient associated with xe, e ∈ δ(q) and yq are equal to 0.

1. Indeed, there exists a solution (x, y) in Pxy(G) lying on the face defined by αx−βy ≤ γwith xuv = 0. This vector can be extended to (x′, y′) in Pxy(G′) where V ′ := V ∪{q, w}and E := E ∪ δ(w) by setting x′e := xe for e ∈ E, y′i := yi for i ∈ V and y′q := 1,x′, y′ := 0 otherwise. This proves that β ′

q = 0.

2. For each z ∈ V , there exists a solution (x, y) in Pxy(G) such that αx − βy = γwith xzl = 1, l ∈ Vh(v)\{v} if z 6∈ Vh(v), l ∈ Vh(u)\{u} otherwise. This vector canbe extended to (x′, y′) in Pxy(G′) where V ′ := V ∪ {q, w} and E ′ := E′ ∪ {{z, q}}with E′ initialized to E ∪ δ(w) by setting x′e := xe for e ∈ E, y′i := yi for i ∈ V andx′zq = y′q := 1, x′, y′ := 0 otherwise. This shows that α′

zq = 0 for all z ∈ V .

Notice from the proof of Theorem 12 that the result also holds in case Vk+1 = {w} is addedinstead of Vk+1 = {w, q}.

These simple lifting results enable us to simplify what follows, since, when proving inequal-ities to be valid or facet-inducing, we only have to consider inequalities which do not arise fromsimpler inequalities by lifting. These simplest inequalities will be called primitive.

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1 2 3 4 5

6 7 8 9

1

2

3

4

5 6 7 8

Figure 13: Two configurations of edges leading to 5-holes in GC .

6.3 Holes, cycles and rings

In Subsection 6.1 we noted that the star and 3-cycle inequalities are equivalent to the well-knownclique inequalities, defined by Padberg (1973) for the stable set polytope. Another well-knownclass of inequalities for the stable set polytope, also due to Padberg (1973), are the so-calledodd hole inequalities. An odd hole in a graph is a chordless cycle with cardinality odd and atleast 5. The associated odd hole inequality states that, if an odd hole is of cardinality c, thenat most bc/2c vertices in the hole can appear in a stable set. Odd hole inequalities can be liftedto obtain facets of the stable set polytope (Padberg (1973)).

In the context of the GSP, the odd hole inequalities state that, if a set F of edges with|F | ≥ 5 and |F | odd induces a hole in the conflict graph GC , then x(F ) ≤ b|F |/2c is valid forPx(G) (and therefore for Pxy(G) also).

It should be immediately apparent that the primitive odd cycle inequalities (i.e. with |Vk| =2) are odd hole inequalities and that non-primitive odd cycle inequalities are lifted odd holeinequalities. However, there are other configurations of edges in G which lead to odd holes inGC , which can be used to derive new facets for Px(G).

Consider the two configurations shown in Figure 13. For the instance on the left in Figure13, the odd hole inequality x16 + x23 + x45 + x39 + x78 ≤ 2 is valid. Similarly, for the instanceon the right in Figure 13, the odd hole inequality x13 +x15 +x24 +x48 +x67 ≤ 2 is valid. Theseinequalities, which happen to be primitive, are not odd cycle inequalities; yet they can be easilyshown to be facet-inducing for the associate polytope Px(G). (As we point out below, however,they are not facet-inducing for Pxy(G)!)

It is not difficult to characterize the possible configurations of edges in G which lead toodd holes in GC . However, we do not go into detail here, because we have not found it to beparticularly illuminating. Moreover, it appears to be difficult to find a closed-form for the liftedodd hole inequalities which induce facets of Px(G). Of course, given any facet-inducing odd holeinequality, we can apply vertex-cloning to obtain more general lifted odd hole inequalities. Butthere are lifted odd hole inequalities which cannot be obtained in this way. Indeed, inequalitiesobtained by vertex-cloning odd hole inequalities have coefficients of zero or one on the left handside — yet we know of lifted odd hole inequalities in which one or more left hand side coefficientstake the value two.

We now turn our attention to the polytope Pxy(G). Paradoxically, the situation here is muchsimpler. It turns out that the only odd hole inequalities which induce facets of Pxy(G) are thosewhich happen to be odd cycle inequalities. This is because ‘weird’ odd hole inequalities suchas the ones shown in Figure 13 are dominated by a much simpler class of inequalities involvingboth x and y variables. This new class of inequalities is introduced in the next two theorems.

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First we will need a definition.

Definition 2 (Rings). Consider a GSP instance with |K| = m ≥ 3 such that |Vk| ≤ 2 for allk ∈ K. A ring is a set F ⊂ E such that:

• |F | = |K|,

• |F ∩ δ(Vk)| = 2 for all k ∈ K and

• F induces a Hamiltonian cycle in the shrunk graph.

Note that for a given ring, there are three possibilities for a cluster. If the cluster containstwo vertices, either each of the two vertices in the cluster is adjacent to an edge in the ring, orone of the vertices in the cluster is adjacent to two edges in the ring and the other vertex inthe cluster is adjacent to none. If the cluster contains one vertex, this vertex is adjacent to twoedges in the ring. We will say that clusters with two vertices of degree one in the ring are odd,whereas clusters of the other types are even.

Theorem 13 (Primitive Odd Ring Inequalities). Let G be such that |Vk| ≤ 2 for all k ∈ Kand let F be a ring. Suppose there are c odd clusters, with c odd, and let W denote the set ofvertices in even clusters which are incident on two edges in the ring. (Note that c+ |W | = m.)Then the (primitive) odd ring inequality

x(F ) ≤ bc/2c+ y(W ) (21)

is valid for Pxy(G).

Proof. For each edge e = {u, v} ∈ F , the inequalities xe ≤ yu and xe ≤ yv hold. Sum thesetogether over all e ∈ F , together with the inequalities (2) for each odd cluster, to obtain

2x(F ) ≤ c+ 2y(W ).

Divide this by two and round down the right hand side to obtain the result.

Applying trivial lifting and vertex-cloning to the primitive inequalities (21) yields what wecall odd ring inequalities.

Theorem 14 (Odd Ring Inequalities). Odd ring inequalities induce facets of Pxy(G).

Proof. This needs to be proved for the primitive version. The result then follows from Propo-sitions 11 and 13. The primitive odd ring inequalities with c ≥ 3 are facet-defining inequalitiessince they are obtained from odd cycle inequalities using the lifting Theorem 12. To prove thatodd ring inequalities with c = 1 are also facet-defining, it suffices to show that the primitiveinequality involving three clusters (one odd and two even clusters) is facet-defining. This can bedone by listing affinely independent vectors lying in the face. Theorem 12 can then be applied tohave the facet-defining property for any primitive odd ring inequality with one odd cluster.

It should be noted that odd ring inequalities are a proper generalization of odd cycle in-equalities. Moreover, as stated above, the odd ring inequalities dominate ‘wierd’ odd holeinequalities such as the ones shown in Figure 13. Indeed, for the instance on the left in Fig-ure 13, the odd hole inequality x16 + x23 + x45 + x39 + x78 ≤ 2 is dominated by the oddring inequality x16 + x23 + x39 + x78 ≤ 1 + y3 (together with the trivial inequalities x45 ≤ y4

and y3 + y4 ≤ 1). Similarly, for the instance on the right in Figure 13, the odd hole inequalityx13+x15+x24+x48+x67 ≤ 2 is dominated by the odd ring inequality x13+x15+x48+x67 ≤ 1+y1

(together with the trivial inequalities x24 ≤ y2 and y1 + y2 ≤ 1).

24

6.4 On odd clique matching inequalities

We now return to the question of which OCMIs (7) are facet-inducing. In order to do this, it isuseful to note that the OCMIs are analogous to the well-known blossom inequalities of Edmonds(1965a). The blossom inequalities state that, if S is a set of vertices in a graph, |S| is odd,and E(S) is the set of all edges which have both end-vertices in S, then a (1-)matching canuse at most bS/2c edges in E(S). Looking at Figure 5 in Section 2, it is clear that the OCMIshave much the same flavour, except that each vertex in S has become a cluster partitioned in acertain way.

This suggests that we might be able to ‘borrow’ the conditions for a blossom inequality to befacet-defining for the matching polytope (Pulleyblank and Edmonds (1974)), and apply them tothe OCMIs. These conditions are based on the concepts of biconnectedness and hypomatchability.We have defined biconnectedness above. A graph G = (V,E) is hypomatchable if, for each vertexi, the subgraph induced in G by V \ {i} possesses a perfect matching. Obviously this impliesthat |V | is odd.This leads to the following two results (which are true for both Pxy(G) and Px(G)):

Theorem 15. If an OCMI is facet-inducing, then its associated shrunk support graph must bebiconnected.

Proof. If the shrunk graph GS(V S , ES) (with |V S | odd), is not biconnected, there exists at leastone articulation point and it can be decomposed into blocks. Two cases arise.

1. At least one leaf block is odd (see Figure 14 (a)). It is denoted by B(V SB , ES

B) with |V SB |

odd. Let k be the articulation point in V SB . The OCMI is the sum of

OCM(B) +OCM(GS\(V SB \{k})) ≤

|V SB | − 1

2+|V S\V S

B |

2=|V S | − 1

2,

where OCM(H) denotes the left hand side of the OCMI whose shrunk support graph isH.

2. All the leaf blocks are even (see Figure 14 (b)). Let B be a leaf block and k be thearticulation point in V S

B . Then the OCMI is the sum of

OCM(B\{k}) +OCM(GS\V SB ) + star(k) ≤

|V SB \{k}| − 1

2+|V S\V S

B | − 1

2+ 1 =

|V S | − 1

2,

where star(k) is the left hand side of the star inequality (12) (which is still valid forPxy

GS(G)) whose nucleus is Vk in G and such that its shrunk support graph is included inGS .

Theorem 16. If an OCMI is facet-inducing, then its associated shrunk support graph must behypomatchable.

We do not present the proof for this theorem, because we will now prove something stronger.It turns out that these two conditions are not sufficient to get a facet. Indeed, Figure 15 (a)shows an OCMI whose shrunk support graph is biconnected and hypomatchable, but which is

25

kk

GS \ V SB B

BGS \ (V S

B \ {k})

(a) If at least one leaf block is odd. (b) If all the leaf blocks are even.

Figure 14: Two cases for the proof of Theorem 15.

(a) OCMI with shrunk graph biconnected and hypomatchable. (b) A stronger lifted odd hole inequality.

Figure 15: An OCMI satisfying the conditions of Theorems 15 and 16, but not facet inducing.

not facet-defining. It is dominated by the inequality shown in Figure 15 (b), which is a liftedodd hole inequality. The right hand side of both inequalities is 2.

In order to give a stronger necessary condition for an OCMI to induce a facet, we will needthe following definitions.

Definition 3. Let G = (V,E) be a graph. A set S = {u, v, w} ⊆ V such that |S| = 3 is said tobe a fork if |E({u, v, w})| ≥ 2.

Definition 4. A graph G = (V,E) is said to be fork-hypomatchable if, for every fork S ⊆ V ,the graph induced by V \ S contains a perfect matching.

Note that the shrunk support graph for the OCMI shown in Figure 15 (a) is biconnectedand hypomatchable, but not fork-hypomatchable.

Theorem 17. If an OCMI is facet-inducing, then its associated shrunk support graph must befork-hypomatchable.

Proof. Consider the OCMI (7) once more. Suppose without loss of generality that V 11 , V

12 , V

22 , V

23

are all non-empty. That is, the clusters V1, V2 and V3 correspond to a fork {u, v, w} in theshrunk graph. Suppose also that removing {u, v, w} from the shrunk graph leaves a graph with

26

Figure 16: Two graphs which satisfy the conditions of Conjecture 3.

no perfect matching. Then (7) is dominated by the following inequality,

x(E(V 11 : V 2

2 )) + x(E(V 12 : V 2

3 )) +c∑

i=1

c−1∑

j=i

x(E(V ji : V i

j+1)) ≤ bc/2c,

which is easily shown to be valid.

Note that, if a graph is biconnected, then every vertex is contained in at least one fork. There-fore, biconnectedness and fork-hypomatchability collectively imply hypomatchability. ThereforeTheorems 15 and 17 imply Theorem 16. Moreover, we propose the following conjecture.

Conjecture 2. An OCMI is facet-inducing (for both Pxy(G) and Px(G)) if and only if itsassociated shrunk support graph is biconnected and fork-hypomatchable.

Enumeration shows that the unique graphs with 5 vertices which satisfy this condition arethe 5-hole and the 5-clique. However, for seven or more vertices there are graphs which meetthe conditions which are neither odd holes nor odd cliques. Figure 16 shows two examples. Thesecond is of particular interest, because it is not Hamiltonian. This implies that the associatedOCMIs are not lifted odd hole inequalities.

It is important to note that, unlike ordinary hypomatchability, fork-hypomatchability is notpreserved under the addition of edges.

6.5 {0, 12}-cuts

It is instructive to note that many of the inequalities discussed in this paper can be derived asso-called {0, 1

2}-cuts. These are valid inequalities which (see Caprara and Fischetti (1996)) areobtained by summing together a number of other valid inequalities, dividing the result by twoand rounding down coefficients to the nearest integer.

In the case of Px(G), it is easy to show that the odd hole inequalities from the conflictgraph can be derived as {0, 1

2}-cuts from the inequalities (10) mentioned at the beginning of thissection. Moreover, since the star inequalities (12) dominate the inequalities (10), we can expectto derive stronger {0, 1

2}-cuts from them. Indeed, many (but not all) lifted odd hole inequalitiescan be derived as {0, 1

2}-cuts from the star inequalities, and so can all of the OCMIs.In general, we expect to be able to derive stronger {0, 1

2}-cuts for Pxy(G) than we can forPx(G), because for Pxy(G) we can use the inequalities (2) and (3), whereas for Px(G) we arelimited to the weaker star inequalities. Indeed, in the case of Pxy(G), the {0, 1

2}-cuts includenot only certain lifted odd hole inequalites and all OCMIs, but also the odd ring inequalities(see the proof of Theorem 13).

27

In fact, in the case of Pxy(G) at least, there exist non-redundant {0, 12}-cuts which do not

lie in any of the classes discussed so far.

Proposition 15. Suppose that |Vk| > 1 for all k. Label each cluster as type 1 or type 2. LetF ⊂ E be a set of edges with the following property:

• If Vk is a cluster of type 1, then for each v ∈ Vk, |F ∩ δ(v)| = 1.

• If Vk is a cluster of type 2, then for each v ∈ Vk, |F ∩ δ(v)| = 2.

Suppose there are t1 clusters of type 1 and that t1 is odd. Finally, let W ⊂ K be the set ofclusters of type 2. Then the (primitive) inequality

x(F ) ≤ bt1/2c+∑

k∈W

y(Vk) (22)

is a valid {0, 12}-cut for P

xy(G).

Proof. For each edge e = {u, v} ∈ F , the inequalities xe ≤ yu and xe ≤ yv hold. Sum thesetogether over all e ∈ F , together with the inequalities (2) for each cluster of type 1, to obtain

2x(F ) ≤ t1 + 2∑

k∈W

y(Vk).

Dividing this by two and rounding down the right hand side yields the inequality (22), thusshowing that (22) is a valid {0, 1

2}-cut.

These {0, 12}-cuts can in fact be obtained by applying Theorem 12 to the OCMIs described

in Proposition 2. Thus, they will induce facets whenever the ‘underlying’ OCMI does.It seems that a further study of {0, 1

2}-cuts for the GSP might be useful. Moreover, in thecontext of a practical cutting plane algorithm for the GSP, it may be more sensible to searchfor violated {0, 1

2}-cuts, rather than limiting the search to predefined classes such as OCMIs.

7 Conclusions

In this paper we have explored the GSP from a theoretical point of view. We have studied itscomplexity and approximability, and have provided new polyhedral results.

It should be stressed that the polyhedral results given in this paper have implications forother problems related to the GSP. All of the valid inequalities for the GSP are also valid forany of the problems which are a restriction of the GSP, such as the GMSTP and GTSP. Ofcourse, the inequalities are no longer guaranteed to induce facets for these other problems. Butin many important cases they do. For example, many of the odd clique matching inequalitiesare facet-inducing for the GMSTP.

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