Annealed replication: a new heuristic for the maximum clique problem

Abstract

We propose a new heuristic for approximating the maximum clique problem based on a detailed analysis of a class of continuous optimization models which provide a complete characterization of solutions to this NP-hard combinatorial problem. We start from a known continuous formulation of the maximum clique, and tackle the search for local solutions with replicator dynamics, a class of dynamical systems developed in various branches of mathematical biology. Hereby, we add to the objective used in previous works a regularization term that controls the global shape of the energy landscape, that is the function actually maximized by the dynamics. The parameter controlling the regularization is changed during the evolution of the dynamical system to render inefficient local solutions (which formerly were stable) unstable, thus conducting the system to escape from sub-optimal points, and so to improve the final results. The role of this parameter is thus superficially similar to that of temperature in simulated annealing in the sense that its variation allows to find better solutions for the problem at hand. We demonstrate several theoretical results on the regularization term and we further support the validity of this approach, reporting on its performances when applied to selected DIMACS benchmark graphs.