{"title":"Achieving Tight \\(O(4^k)\\) Runtime Bounds on Jumpk by Proving that Genetic Algorithms Evolve Near-Maximal Population Diversity","authors":"Andre Opris, Johannes Lengler, Dirk Sudholt","doi":"10.1007/s00453-025-01323-x","DOIUrl":null,"url":null,"abstract":"<div><p>The <span>\\(\\textsc {Jump} _k\\)</span> benchmark was the first problem for which crossover was proven to give a speed-up over mutation-only evolutionary algorithms. Jansen and Wegener (Algorithmica 2002) proved an upper bound of <span>\\(O(\\textrm{poly}(n) + 4^k/p_c)\\)</span> for the (<span>\\(\\mu \\)</span>+1) Genetic Algorithm ((<span>\\(\\mu \\)</span>+1) GA), but only for unrealistically small crossover probabilities <span>\\(p_c\\)</span>. To this date, it remains an open problem to prove similar upper bounds for realistic <span>\\(p_c\\)</span>; the best known runtime bound, in terms of function evaluations, for <span>\\(p_c = \\Omega (1)\\)</span> is <span>\\(O((n/\\chi )^{k-1})\\)</span>, <span>\\(\\chi \\)</span> a positive constant. We provide a novel approach and analyse the evolution of the population diversity, measured as sum of pairwise Hamming distances, for a variant of the (<span>\\(\\mu \\)</span>+1) GA on <span>\\(\\textsc {Jump} _k\\)</span>. The (<span>\\(\\mu \\)</span>+1)-<span>\\({\\lambda _c}\\)</span>-GA creates one offspring in each generation either by applying mutation to one parent or by applying crossover <span>\\({\\lambda _c}\\)</span> times to the same two parents (followed by mutation), to amplify the probability of creating an accepted offspring in generations with crossover. We show that population diversity in the (<span>\\(\\mu \\)</span>+1)-<span>\\({\\lambda _c}\\)</span>-GA converges to an equilibrium of near-perfect diversity. This yields an improved time bound of <span>\\(O(\\mu n \\log (\\mu ) + 4^k)\\)</span> function evaluations for a range of <i>k</i> under the mild assumptions <span>\\(p_c = O(1/k)\\)</span> and <span>\\(\\mu \\in \\Omega (kn)\\)</span>. For all constant <i>k</i>, the restriction is satisfied for some <span>\\(p_c = \\Omega (1)\\)</span> and it implies that the expected runtime for all constant <i>k</i> and an appropriate <span>\\(\\mu = \\Theta (kn)\\)</span> is bounded by <span>\\(O(n^2 \\log n)\\)</span>, irrespective of <i>k</i>. For larger <i>k</i>, the expected time of the (<span>\\(\\mu \\)</span>+1)-<span>\\({\\lambda _c}\\)</span>-GA is <span>\\(\\Theta (4^k)\\)</span>, which is tight for a large class of unbiased black-box algorithms and faster than the original (<span>\\(\\mu \\)</span>+1) GA by a factor of <span>\\(\\Omega (1/p_c)\\)</span>. We also show that our analysis can be extended to other unitation functions such as <span>\\(\\textsc {Jump} _{k, \\delta }\\)</span> and H<span>urdle</span>.</p></div>","PeriodicalId":50824,"journal":{"name":"Algorithmica","volume":"87 11","pages":"1564 - 1619"},"PeriodicalIF":0.7000,"publicationDate":"2025-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00453-025-01323-x.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Algorithmica","FirstCategoryId":"94","ListUrlMain":"https://link.springer.com/article/10.1007/s00453-025-01323-x","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"COMPUTER SCIENCE, SOFTWARE ENGINEERING","Score":null,"Total":0}
引用次数: 0
Abstract
The \(\textsc {Jump} _k\) benchmark was the first problem for which crossover was proven to give a speed-up over mutation-only evolutionary algorithms. Jansen and Wegener (Algorithmica 2002) proved an upper bound of \(O(\textrm{poly}(n) + 4^k/p_c)\) 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, in terms of function evaluations, for \(p_c = \Omega (1)\) is \(O((n/\chi )^{k-1})\), \(\chi \) a positive constant. We provide a novel approach and analyse the evolution of the population diversity, measured as sum of pairwise Hamming distances, for a variant of the (\(\mu \)+1) GA on \(\textsc {Jump} _k\). The (\(\mu \)+1)-\({\lambda _c}\)-GA creates one offspring in each generation either by applying mutation to one parent or by applying crossover \({\lambda _c}\) times to the same two parents (followed by mutation), to amplify the probability of creating an accepted offspring in generations with crossover. We show that population diversity in the (\(\mu \)+1)-\({\lambda _c}\)-GA converges to an equilibrium of near-perfect diversity. This yields an improved time bound of \(O(\mu n \log (\mu ) + 4^k)\) function evaluations 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 \(p_c = \Omega (1)\) and it implies that the expected runtime for all constant k and an appropriate \(\mu = \Theta (kn)\) is bounded by \(O(n^2 \log n)\), irrespective of k. For larger k, the expected time of the (\(\mu \)+1)-\({\lambda _c}\)-GA is \(\Theta (4^k)\), which is tight for a large class of unbiased black-box algorithms and faster than the original (\(\mu \)+1) GA by a factor of \(\Omega (1/p_c)\). We also show that our analysis can be extended to other unitation functions such as \(\textsc {Jump} _{k, \delta }\) and Hurdle.
期刊介绍:
Algorithmica is an international journal which publishes theoretical papers on algorithms that address problems arising in practical areas, and experimental papers of general appeal for practical importance or techniques. The development of algorithms is an integral part of computer science. The increasing complexity and scope of computer applications makes the design of efficient algorithms essential.
Algorithmica covers algorithms in applied areas such as: VLSI, distributed computing, parallel processing, automated design, robotics, graphics, data base design, software tools, as well as algorithms in fundamental areas such as sorting, searching, data structures, computational geometry, and linear programming.
In addition, the journal features two special sections: Application Experience, presenting findings obtained from applications of theoretical results to practical situations, and Problems, offering short papers presenting problems on selected topics of computer science.