This paper explores the enhancement of solution diversity in evolutionary algorithms (EAs) for the maximum matching problem, concentrating on complete bipartite graphs and paths. We adopt binary string encoding for matchings and use Hamming distance to measure diversity, aiming for its maximization. Our study centers on the $(\mu+1)$-EA and $2P-EA_D$, which are applied to optimize diversity. We provide a rigorous theoretical and empirical analysis of these algorithms. For complete bipartite graphs, our runtime analysis shows that, with a reasonably small $\mu$, the $(\mu+1)$-EA achieves maximal diversity with an expected runtime of $O(\mu^2 m^4 \log(m))$ for the small gap case (where the population size $\mu$ is less than the difference in the sizes of the bipartite partitions) and $O(\mu^2 m^2 \log(m))$ otherwise. For paths, we establish an upper runtime bound of $O(\mu^3 m^3)$. The $2P-EA_D$ displays stronger performance, with bounds of $O(\mu^2 m^2 \log(m))$ for the small gap case, $O(\mu^2 n^2 \log(n))$ otherwise, and $O(\mu^3 m^2)$ for paths. Here, $n$ represents the total number of vertices and $m$ the number of edges. Our empirical studies, which examine the scaling behavior with respect to $m$ and $\mu$, complement these theoretical insights and suggest potential for further refinement of the runtime bounds.

翻译：本文探讨了针对最大匹配问题的进化算法中解多样性的增强方法，重点研究完全二分图与路径。我们采用二进制串编码表示匹配，并使用汉明距离度量多样性，旨在实现其最大化。研究聚焦于$(\mu+1)$-EA与$2P-EA_D$两种算法在多样性优化中的应用。通过严格的理论与实证分析，对于完全二分图，我们的运行时间分析表明：当种群规模$\mu$合理较小时，在小间隙情形（即种群规模$\mu$小于二分图两部分顶点数量之差）下，$(\mu+1)$-EA以$O(\mu^2 m^4 \log(m))$的期望运行时间达到最大多样性；其他情形则为$O(\mu^2 m^2 \log(m))$。对于路径，我们建立了$O(\mu^3 m^3)$的上界。$2P-EA_D$展现出更优性能：小间隙情形下界为$O(\mu^2 m^2 \log(m))$，其他情形为$O(\mu^2 n^2 \log(n))$，路径情形为$O(\mu^3 m^2)$。其中$n$表示顶点总数，$m$表示边数。我们的实证研究通过考察$m$与$\mu$的标度行为，补充了理论分析，并揭示了进一步优化运行时间上界的潜在方向。