The Maximum Common Subgraph (MCS) problem plays a crucial role across various domains, bridging theoretical exploration and practical applications in fields like bioinformatics and social network analysis. Despite its wide applicability, MCS is notoriously challenging and is classified as an NP-Complete (NPC) problem. This study introduces new heuristics aimed at mitigating these challenges through the reformulation of the MCS problem as the Maximum Clique and its complement, the Maximum Independent Set. Our first heuristic leverages the Motzkin-Straus theorem to reformulate the Maximum Clique Problem as a constrained optimization problem, continuing the work of Pelillo in Replicator Equations, Maximal Cliques, and Graph Isomorphism (1999) with replicator dynamics and introducing annealed imitation heuristics as in Dominant Sets and Hierarchical Clustering (Pavan and Pelillo, 2003) to improve chances of convergence to better local optima. The second technique applies heuristics drawn upon strategies for the Maximum Independent Set problem to efficiently reduce graph sizes as used by Akiwa and Iwata in 2014. This enables faster computation and, in many instances, yields near-optimal solutions. Furthermore we look at the implementation of both techniques in a single algorithm and find that it is a promising approach. Our techniques were tested on randomly generated Erd\H{o}s-R\'enyi graph pairs. Results indicate the potential for application and substantial impact on future research directions.

翻译：最大公共子图（MCS）问题在多个领域扮演着关键角色，连接了生物信息学和社会网络分析等领域的理论探索与实际应用。尽管具有广泛适用性，MCS问题却极具挑战性，被归类为NP完全问题。本研究引入新的启发式方法，通过将MCS问题重新表述为最大团问题及其对偶的最大独立集问题来缓解这些挑战。第一种启发式方法利用Motzkin-Straus定理将最大团问题转化为约束优化问题，延续了Pelillo在《复制方程、极大团与图同构》（1999年）中关于复制动力学的研究，并引入模拟退火模仿启发式方法（如Pavan与Pelillo在《主导集与层次聚类》（2003年）中所述）以提升收敛至更优局部最优解的概率。第二种技术借鉴针对最大独立集问题的启发式策略（如Akiwa与Iwata在2014年提出的方法），通过有效缩减图的规模，实现更快速的求解，并在多数情况下获得接近最优的解。此外，我们将这两种技术整合于单一算法中，发现该方法具有良好前景。本研究的算法在随机生成的Erdős–Rényi图对上进行了测试，结果表明其具有应用潜力，并对未来研究方向产生重要影响。