A Pivoting-Based Heuristic for the Maximum Clique Problem

Abstract

We tackle the maximum clique problem by reformulating it in terms of a standard quadratic program, which derives from a theorem of Motzkin and Straus. A heuristic is then obtained by solving with Lemke's method the linear complementarity problem that arises from the KKT conditions of the QP. We have added a pivoting rule that exploits de-generacy to reach good suboptimal solutions. Very positive results have been obtained on the DIMACS benchmark graphs. A clique is a subset of V in which all vertices are pair-wise adjacent. A clique S is called maximal if no strict superset of S is a clique. A maximum cardinality clique (or, simply, a maximum clique) is a clique whose cardinality is the largest possible. The maximum clique problem 383 N. Hadjisavvas and P.M. Pardalos (eds.), Advances in Convex Analysis and Global Optimization, 383-394. (MAX-CLIQUE) is the problem of finding a maximum clique 8 in G (see [2] for a review). The maximum size of a clique in G is called the clique number (of G) and is typically denoted by w (G). In 1965, Motzkin and Straus [6] established a remarkable connection between MAX-CLIQUE and a certain quadratic problem. Let G = (V, E) be an undirected graph, and let A denote the standard simplex in the n-dimensional Euclidean space lR n: