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.

翻译：最大团问题（MCP）是图论和计算复杂性理论中的一个基本问题。给定一个图G，该问题旨在找出G中最大的团（完全子图）。MCP在不同领域具有许多重要应用，并已得到广泛研究。该问题已被证明是NP-Hard问题，其对应的判定问题是NP-Complete问题。迄今为止发现的所有精确（最优）算法均具有指数时间复杂度。多种元启发式算法已被用于近似求解MCP，包括遗传与模因算法、蚁群优化算法、贪心算法、禁忌搜索算法以及模拟退火算法。本研究批判性地考察了遗传算法（GAs）应用于MCP时相较于纯随机方法的有效性。我们的结果表明，采用随机搜索生成子图并将其优化为团的蒙特卡洛算法，在速度和求解能力上往往优于遗传算法，尤其在稀疏图中表现更为突出。这一发现挑战了传统对遗传算法的依赖，提示需要重新评估交叉算子和变异算子在解空间探索中的作用。我们观察到，在某些较稠密的图中，遗传算法的重组策略展现出意料之外的效能，暗示了遗传方法在特定条件下尚未开发的潜力。这项工作不仅质疑了现有范式，还为探索求解MCP及其他NP-Hard问题的算法效率开辟了新途径，鼓励进一步研究在何种条件下纯随机方法优于遗传重组，反之亦然。