Probabilistic Cellular Automata Monte Carlo for the Maximum Clique Problem

Abstract

We consider the problem of finding the largest clique of a graph. This is an NP-hard problem and no exact algorithm to solve it exactly in polynomial time is known to exist. Several heuristic approaches have been proposed to find approximate solutions. Markov Chain Monte Carlo is one of these. In the context of Markov Chain Monte Carlo, we present a class of “parallel dynamics”, known as Probabilistic Cellular Automata, which can be used in place of the more standard choice of sequential “single spin flip” to sample from a probability distribution concentrated on the largest cliques of the graph. We perform a numerical comparison between the two classes of chains both in terms of the quality of the solution and in terms of computational time. We show that the parallel dynamics are considerably faster than the sequential ones while providing solutions of comparable quality.

Keywords

• maximum clique problem
• probabilistic cellular automata
• Markov chain Monte Carlo
• QUBO
• parallel computing