Replicator Dynamics for Evolution Towards the Maximum Clique: Variations and Experiments*

In this note we report on further investigations about the performance of evolutionary dynamics methods to approximate the maximum clique in an undirected graph. This approach makes use of a continuous characterization of the discrete problem, reformulating it into the question of maximizing a quadratic form over the standard simplex. Frequently, the local optimization part is solved by following the (convergent) trajectories under the so-called replicator dynamics, a versatile tool in biological modelling, along which the objective function always increases. It can be shown that in the limit one obtains always a characteristic vector associated with a maximal clique, and empirical evidence suggests that the quality of these are quite satisfactory compared to other continuous-based heuristics. However, the question remains whether some systematic preprocessing could help to avoid being trapped in the domain of attraction of too small cliques. While previous attacks concentrated (a) on a discrete-time version of the replicator dynamics corresponding to an endogenously specified step size; and (b) starting from the barycentre of the standard simplex, we tried some variants here: we chose the starting point with different, more sophisticated strategies; and we used Runge/Kutta discretization of the continuous time version of replicator dynamics, which is known for its efficiency in theory. The results obtained on selected DIMACS benchmark graphs are reported here, but the picture is is not at all clearly in favor of these variants, suggesting that some global optimization procedures like the recently proposed G.E.N.F. approach (shortly reviewed here) are indeed indispensible to get improved results also for hard instances.