The edge-weighted clique problem: Valid inequalities, facets and polyhedral computations

Abstract

Let K_n=(V,E) be the complete undirected graph with weights c_e associated to the edges in E. We consider the problem of finding the subclique C=(U,F) of K_n such that the sum of the weights of the edges in F is maximized and |U|⩽b, for some b∈[1,…,n]. This problem is called the Maximum Edge-Weighted Clique Problem (MEWCP) and is NP-hard. In this paper we investigate the facial structure of the polytope associated to the MEWCP introducing new classes of facet defining inequalities. Computational experiments with a branch-and-cut algorithm are reported confirming the strength of these inequalities. All instances with up to 48 nodes could be solved without entering into the branching phase. Moreover, we show that some of these new inequalities also define facets of the Boolean Quadric Polytope and generalize many of the previously known inequalities for this well-studied polytope.

Introduction

Let K_n=(V,E) be the complete undirected graph with weights c_e associated to the edges in E. We consider the problem of finding the subclique C=(U,F) of K_n such that the sum of the weights of the edges in F is maximized and |U|⩽b, for some integer b∈[1,…,n]. This problem is called the Maximum Edge-Weighted Clique Problem (MEWCP).

The MEWCP can be easily seen to be NP-hard, since the usual MAX-CLIQUE problem reduces polynomially to it. Heuristic algorithms based on local search have been proposed in [14] to find good suboptimal solutions for this problem.

Exact algorithms based on Integer Programming formulations have been proposed in [5], [6], [12]. The natural formulation presented in [5] uses only binary variables corresponding to the edges of K_n. The authors investigate the problem from a polyhedral point of view. Several facet defining inequalities are introduced and computational results obtained by a cutting-plane algorithm using these inequalities are reported. From their computational experiments, the authors conclude that the cutting-plane approach was not suitable to solve the MEWCP even for moderate sized instances. The largest instance they solve refers to a graph on 25 nodes but extremely poor performances are reported for quite smaller instances.

In [6], an extended formulation is proposed that includes binary variables not only for the edges but also for the nodes in K_n. The authors did not investigate the new model from a polyhedral point of view. A first polyhedral investigation of the extended formulation is done in [12] where several classes of facet defining inequalities for the associated polytope are presented. The authors also prove that the lower bounds provided by the extended formulation are better than those coming from the natural formulation on the edge variables. The computational results reported in [12] are much more encouraging than those reported in [5]. The instances tested include graphs with up to 30 nodes and most of them have been solved to optimally by pure cutting-planes (no branching was necessary).

Many facet defining inequalities introduced in [12] for the MEWCP, such as the clique and cut inequality, are based upon facet defining inequalities for the Boolean Quadric Polytope (BQP) investigated in [11]. In fact, the polytope associated to the extended formulation of MEWCP is contained in the BQP and, therefore, any inequality valid for the BQP is also valid for the polytope associated to MEWCP.

In this paper we go further in investigating the facial structure of the polytope associated to the extended formulation of the MEWCP in order to have a better understanding of it. For this, we introduce three new classes of valid and facet defining inequalities for this polytope. The first class of inequalities proposed here generalizes the clique, the cut and the (s,t)-cut inequalities studied in [11] for the BQP. We prove the validity of these new inequalities and derive sufficient conditions under which they are facet defining for the MEWCP polytope. Moreover, we show that the validity and facetness properties can also be extended to the BQP.

The second and third classes of inequalities generalize the tree inequalities originally introduced in [9] and further studied in [12]. The generalization goes in two different directions. For both of them we have been able to proof that some special cases correspond to facet defining inequalities for the MEWCP.

Besides the search for new classes of facet defining inequalities, we also have carried out computational experiments with a branch-and-cut algorithm that we have implemented. The main goals with these experiments were to evaluate the strength of the new inequalities introduced here and to compare different cutting-plane strategies for the MEWCP. To date, to the best of our knowledge, no comparison between different cutting strategies have appeared in the literature for this problem. We have run the code on instances with up to 48 nodes and the results confirm that at least the first class of inequalities we have introduced is computationally useful. Even when using a pure cutting plane algorithm based on a quite small subset of these inequalities, we were able to solve all instances of our data set to optimality.

The paper is organized as follows. In Section 2 we give the extended Integer Programming formulation for the MEWCP and summarize the main polyhedral results from the literature which are important for our work. Section 3 describes our first class of inequalities generalizing the clique, the cut and the (s,t)-cut inequalities. Section 4 discusses two possible generalizations of the tree inequalities leading to two distinct classes of valid inequalities for the MEWCP. In Section 5 we describe a branch-and-cut algorithm that uses some of the inequalities introduced in the previous section and report our computational results.

Before going to the next section, we introduce some notation that we use in the sequel.

Let S and T be any two disjoint subsets of nodes in a graph G=(V,E). Then, E(S) denotes the set of all edges in the subgraph induced by S in G. The set of all edges having exactly one end node in S and the other end node in T is denoted by δ(S,T). To make the notation easier, for any node $\text{u∈V,}\text{δ(u)}$ is used to denote the set of edges in δ({u},V−{u}).

If π is a vector in $\text{R}{}^{\mathrm{|V|}}$, we define π(S)=∑_u∈Sπ_u. Analogously, if $\text{γ∈}\text{R}{}^{\mathrm{|E|}}$ and H is a subset of edges in E, we define γ(H)=∑_e∈Hγ_e. To simplify the notation, we also use γ(S) and γ(S,T) to denote ∑_e∈E(S)γ_e and ∑_e∈δ(S,T)γ_e, respectively.