A Tight $O(4^k/p_c)$ Runtime Bound for a ($μ$+1) GA on Jump$_k$ for Realistic Crossover Probabilities

• URL: http://arxiv.org/abs/2404.07061v1
• Date: Wed, 10 Apr 2024 14:50:43 GMT
• Title: A Tight $O(4^k/p_c)$ Runtime Bound for a ($μ$+1) GA on Jump$_k$ for Realistic Crossover Probabilities
• Authors: Andre Opris, Johannes Lengler, Dirk Sudholt,
• Abstract summary: We show that population diversity converges to an equilibrium of near-perfect diversity. Our work partially solves a problem that has been open for more than 20 years.
• Score: 1.8434042562191815
• License: http://creativecommons.org/publicdomain/zero/1.0/
• Abstract: The Jump$_k$ 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 $O({\rm poly}(n) + 4^k/p_c)$ for the ($\mu$+1)~Genetic Algorithm ($(\mu+1)$ GA), but only for unrealistically small crossover probabilities $p_c$. To this date, it remains an open problem to prove similar upper bounds for realistic~$p_c$; the best known runtime bound for $p_c = \Omega(1)$ is $O((n/\chi)^{k-1})$, $\chi$ 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 \muga on Jump$_k$. We show that population diversity converges to an equilibrium of near-perfect diversity. This yields an improved and tight time bound of $O(\mu n \log(k) + 4^k/p_c)$ for a range of~$k$ under the mild assumptions $p_c = O(1/k)$ and $\mu \in \Omega(kn)$. For all constant~$k$ the restriction is satisfied for some $p_c = \Omega(1)$. Our work partially solves a problem that has been open for more than 20 years.

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