More Effective Crossover Operators for the All-Pairs Shortest Path Problem

TL;DR

This paper investigates how different crossover strategies, including repair mechanisms and parent selection, can significantly improve the efficiency of evolutionary algorithms solving the all-pairs shortest path problem.

Contribution

It introduces and analyzes simple modifications to crossover operators that asymptotically enhance the runtime of evolutionary algorithms for the problem.

Findings

Repair mechanisms improve expected optimization time to O(n^{3.2} (log n)^{0.2})

Parent selection for feasible offspring reduces time to O(n^{3} log n)

Simple crossover adjustments can significantly speed up evolutionary algorithms.

Abstract

The all-pairs shortest path problem is the first non-artificial problem for which it was shown that adding crossover can significantly speed up a mutation-only evolutionary algorithm. Recently, the analysis of this algorithm was refined and it was shown to have an expected optimization time (w.r.t. the number of fitness evaluations) of $\Theta(n^{3.25}(\log n)^{0.25})$. In contrast to this simple algorithm, evolutionary algorithms used in practice usually employ refined recombination strategies in order to avoid the creation of infeasible offspring. We study extensions of the basic algorithm by two such concepts which are central in recombination, namely \emph{repair mechanisms} and \emph{parent selection}. We show that repairing infeasible offspring leads to an improved expected optimization time of $\mathord{O}(n^{3.2}(\log n)^{0.2})$. As a second part of our study we prove that choosing parents that guarantee feasible offspring results in an even better optimization time of $\mathord{O}(n^{3}\log n)$. Both results show that already simple adjustments of the recombination operator can asymptotically improve the runtime of evolutionary algorithms.