preprints.org > computer science and mathematics > probability and statistics > doi: 10.20944/preprints202408.2259.v1 (registering DOI)

Preprint Article Version 1 This version is not peer-reviewed

Version 1 : Received: 30 August 2024 / Approved: 30 August 2024 / Online: 30 August 2024 (14:56:57 CEST)

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 being one of those. 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 one while providing solutions of comparable quality.

Maximum Clique Problem; Probabilistic Cellular Automata; Monte Carlo Markov Chain; QUBO; Parallel Computing

Computer Science and Mathematics, Probability and Statistics

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