A Tight O(4k/pc) Runtime Bound for a (μ+1)GA on Jumpk for Realistic Crossover Probabilities
Pages 1605 - 1613
Abstract
The Jumpk benchmark was the first problem for which crossover was proven to give a speedup over mutation-only evolutionary algorithms. Jansen and Wegener (2002) proved an upper bound of O(poly(n) + 4k/pc) for the (μ+1) Genetic Algorithm ((μ+1) GA), but only for unrealistically small crossover probabilities pc. To this date, it remains an open problem to prove similar upper bounds for realistic pc; the best known runtime bound for pc = Ω(1) is O((n/x)k-1), χ a positive constant.
Using recently developed techniques, we analyse the evolution of the population diversity, measured as sum of pairwise Hamming distances, for a variant of the (μ+1) GA on Jumpk. We show that population diversity converges to an equilibrium of near-perfect diversity. This yields an improved and tight time bound of O(μn log(k) + 4k/pc) for a range of k under the mild assumptions pc = O(1/k) and μ ϵ Ω(kn). For all constant k the restriction is satisfied for some pc = Ω(1). Our work partially solves a problem that has been open for more than 20 years.
References
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Andre Opris, Johannes Lengler, and Dirk Sudholt. 2024. A Tight O(4k/pc) Runtime Bound for a (μ+1) GA on Jumpk for Realistic Crossover Probabilities. Technical Report. http://arxiv.org/abs/2404.07061
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Index Terms
- A Tight O(4k/pc) Runtime Bound for a (μ+1)GA on Jumpk for Realistic Crossover Probabilities
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Published: 14 July 2024
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