Abstract
The idea to recombine two or more search points into a new solution is one of the main design principles of evolutionary computation (EC). Its usefulness in the combinatorial optimization context, however, is subject to a highly controversial discussion between EC practitioners and the broader Computer Science research community. While the former, naturally, report significant speedups procured by crossover, the belief that sexual reproduction cannot advance the search for high-quality solutions seems common, for example, amongst theoretical computer scientists. Examples that help understand the role of crossover in combinatorial optimization are needed to promote an intensified discussion on this subject.
We contribute with this work an example of a crossover-based genetic algorithm (GA) that provably outperforms any mutation-based black-box heuristic on a classic benchmark problem. The appeal of our examples lies in its simplicity: the proof of the result uses standard mathematical techniques and can be taught in a basic algorithms lecture.
Our theoretical result is complemented by an empirical evaluation, which demonstrates that the superiority of the GA holds already for quite small problem instances.
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Notes
- 1.
See, for example, here: https://www.simonsfoundation.org/2010/05/18/why-sex/.
- 2.
See Sect. 3 for a discussion of the fact that sampling the parent without replacement improves the expected optimization time of this algorithm on OneMax. We do not apply this modified parent selection rule in the greedy \((\mu +1)\) GA\(_{\text {mod}}\) to highlight that the main improvement stems from the modified mutation step.
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Acknowlegement
We thank the anonymous reviewers of this paper for their constructive feedback, which has helped us to improve the presentation of our main result. This research benefited from the support of the FMJH Program Gaspard Monge in optimization and operation research, and from the support to this program from EDF.
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Pinto, E.C., Doerr, C. (2018). A Simple Proof for the Usefulness of Crossover in Black-Box Optimization. In: Auger, A., Fonseca, C., Lourenço, N., Machado, P., Paquete, L., Whitley, D. (eds) Parallel Problem Solving from Nature – PPSN XV. PPSN 2018. Lecture Notes in Computer Science(), vol 11102. Springer, Cham. https://doi.org/10.1007/978-3-319-99259-4_3
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