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

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 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 repair mechanisms and parent selection. We show that repairing infeasible offspring leads to an improved expected optimization time of 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 O(n 3 log n).Both results show that already simple adjustments of the recombination operator can asymptotically improve the runtime of evolutionary algorithms. * This work greatly benefited from various discussions at the Dagstuhl seminar 08051 on Theory of Evolutionary Algorithms.

More effective crossover operators for the all-pairs shortest path problem

“…This paper is an extension of the conference publication [10]. Here, we present full proofs and additionally discuss our structural insights in the new discussion section (Section 6).…”

“…To derive the two inequalities (10) and (12) in Proposition 13, we apply a result from [9], which we do not prove here but simply state as Lemma 14. We adapt the notation to our needs, a closer look into the proofs in [9] shows that this is easily possible.…”

How crossover helps in pseudo-boolean optimization

“…Speedups by crossover could be shown for coloring problems inspired by the Ising model [8,21] and for the all-pairs shortest path problem [3,5,4,16,11]. In terms of pseudo-Boolean optimization, most results were actually limited to artificial functions [13,22,19,20] constructed such that a theoretical analysis was possible.…”

Running Time Analysis of MOEA/D with Crossover on Discrete Optimization Problem

“…There are several studies have contributed to this issue. For single optimization problems, these researches (Jansen and Wegener 2002;Kötzing, Sudholt, and Theile 2011;Lehre and Yao 2011;Doerr et al 2013;Doerr, Doerr, and Ebel 2015;Sudholt 2017;Doerr and Doerr 2018;Dang et al 2018) have shown that crossover operator can speed up the search. Specifically, they proved that EAs enabled crossover outperform the disabled ones on some benchmark discrete optimization problems in term of the expected running time.…”