How Population Diversity Influences the Efficiency of Crossover

Our theoretical understanding of crossover is limited by our ability to analyze how population diversity evolves. In this study, we provide one of the first rigorous analyses of population diversity and optimization time in a setting where large diversity and large population sizes are required to speed up progress. We give a formal and general criterion which amount of diversity is necessary and sufficient to speed up the \((\mu +1)\) Genetic Algorithm on LeadingOnes. We show that the naturally evolving diversity falls short of giving a substantial speed-up for any \(\mu =O(\sqrt{n}/\log ^2 n)\) . On the other hand, we show that even for \(\mu =2\) , if we simply break ties in favor of diversity then this increases diversity so much that optimization is accelerated by a constant factor.3(Proofs in this submission are mostly omitted due to the page limit. A full version with detailed proofs can be found in the arXiv version of this article [2], but reviewers are not required to consult that version or to check correctness of those proofs.)