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.

An analysis on recombination in multi-objective evolutionary optimization

“…Crossover operators subsequently have been shown to be useful in more cases. These include Ising models [16,49], the TwoPaths instance class of the problem of computing unique input-output sequences [30], some instance classes of the vertex cover problem [39], and the all-pairs shortest path problem [10,9,11]. Meanwhile, on the contrary, Richter et al [44] produced the Ignoble Trail functions where a crossover operator was shown to be harmful.…”

Improved time complexity analysis of the Simple Genetic Algorithm

“…However, these techniques cannot be directly applied to the analyses of more realistic GAs incorporating a crossover operator. Several results were indeed available proving that crossover is useful (Jansen and Wegener, 2005;Watson and Jansen, 2007;Oliveto et al, 2008;Doerr et al, 2010;Kötzing et al, 2011;Neumann et al, 2011;Sudholt, 2012;Doerr et al, 2013), but they rely heavily on elitist selection operators. Moreover, mostly only upper bounds on the running time of crossover-based algorithms were available.…”

Black-box complexity of the binary value function

“…the unary unbiased black-box complexity of OneMax was proven to be Θ(n log n) [6,16]. Together with the rather old question of whether crossover is useful in evolutionary algorithms (which was previously answered positively [8,15,19]), this inspired a number of works on higher-arity unbiased algorithms, since many crossovers are binary unbiased operators.…”