A new trust region technique for the maximum weight clique problem

A new simple generalization of the Motzkin-Straus theorem for the maximum weight clique problem is formulated and directly proved. Within this framework a trust region heuristic is developed. In contrast to usual trust region methods, it regards not only the global optimum of a quadratic objective over a sphere, but also a set of other stationary points of the program. We formulate and prove a condition when a Motzkin-Straus optimum coincides with such a point. The developed method has complexity O(n(3)), where n is the number of vertices of the graph. It was implemented in a publicly available software package QUALEX-MS. Computational experiments indicate that the algorithm is exact on small graphs and very efficient on the DIMACS benchmark graphs and various random maximum weight clique problem instances. (c) 2006 Elsevier B.V. All rights reserved.