Analysis of Evolutionary Diversity Optimisation for the Maximum Matching Problem

Abstract

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.