Approximating the maximum weight clique using replicator dynamics

Given an undirected graph with weights on the vertices, the maximum weight clique problem (MWCP) is to find a subset of mutually adjacent vertices (a clique) having the largest total weight. This is a generalization of the problem of finding the maximum cardinality clique of an unweighted graph, which is the special case of the MWCP when all vertex weights are equal. The problem is NP-hard for arbitrary graphs, and so is the problem of approximating it within a constant factor. We present a parallel, distributed heuristic for approximating the MWCP based on dynamics principles. It centers around a continuous characterization of the MWCP (a purely combinatorial problem), and lets it be formulated in terms of continuous quadratic programming. One drawback is the presence of spurious solutions, and we present their characterizations. To avoid them we introduce a regularized continuous formulation of the MWCP and show how it completely solves the problem. The formulation naturally maps onto a parallel, distributed computational network whose dynamical behavior is governed by the replicator equations. These are dynamical systems introduced in evolutionary game theory and population genetics to model evolutionary processes on a macroscopic scale. We present theoretical results which guarantee that the solutions provided by our clique finding replicator network are actually those sought. Experimental results confirm the effectiveness of the proposed approach.