More Effective Crossover Operators for the All-Pairs Shortest Path Problem
This work addresses runtime improvements for evolutionary algorithms in combinatorial optimization, but it is incremental as it builds on prior analyses of crossover effects.
The paper tackled the all-pairs shortest path problem by extending an evolutionary algorithm with crossover operators, showing that repair mechanisms improve expected optimization time to O(n^{3.2}(log n)^{0.2}) and parent selection for feasible offspring improves it to O(n^{3} log n).
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 $Θ(n^{3.25}(\log n)^{0.25})$. 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.