Recombination vs Stochasticity: A Comparative Study on the Maximum Clique Problem

TL;DR

This paper compares genetic algorithms and Monte Carlo stochastic methods for solving the maximum clique problem, finding that stochastic approaches often outperform genetic algorithms, especially in less dense graphs, challenging traditional assumptions.

Contribution

It provides a critical comparison showing stochastic methods can be more effective than genetic algorithms for MCP, highlighting conditions where each approach excels.

Findings

Monte Carlo algorithms outperform GAs in speed and effectiveness in less dense graphs.

Genetic algorithms show unexpected efficacy in denser graphs due to recombination strategies.

Results suggest reevaluating the reliance on genetic operators for solving NP-hard problems.

Abstract

The maximum clique problem (MCP) is a fundamental problem in graph theory and in computational complexity. Given a graph G, the problem is that of finding the largest clique (complete subgraph) in G. The MCP has many important applications in different domains and has been much studied. The problem has been shown to be NP-Hard and the corresponding decision problem to be NP-Complete. All exact (optimal) algorithms discovered so far run in exponential time. Various meta-heuristics have been used to approximate the MCP. These include genetic and memetic algorithms, ant colony optimization, greedy algorithms, Tabu algorithms, and simulated annealing. This study presents a critical examination of the effectiveness of applying genetic algorithms (GAs) to the MCP compared to a purely stochastic approach. Our results indicate that Monte Carlo algorithms, which employ random searches to generate and then refine sub-graphs into cliques, often surpass genetic algorithms in both speed and capability, particularly in less dense graphs. This observation challenges the conventional reliance on genetic algorithms, suggesting a reevaluation of the roles of the crossover and mutation operators in exploring the solution space. We observe that, in some of the denser graphs, the recombination strategy of genetic algorithms shows unexpected efficacy, hinting at the untapped potential of genetic methods under specific conditions. This work not only questions established paradigms but also opens avenues for exploring algorithmic efficiency in solving the MCP and other NP-Hard problems, inviting further research into the conditions that favor purely stochastic methods over genetic recombination and vice versa.