Towards Analyzing Crossover Operators in Evolutionary Search via General Markov Chain Switching Theorem

Evolutionary algorithms (EAs), simulating the evolution process of natural species, are used to solve optimization problems. Crossover (also called recombination), originated from simulating the chromosome exchange phenomena in zoogamy reproduction, is widely employed in EAs to generate offspring solutions, of which the effectiveness has been examined empirically in applications. However, due to the irregularity of crossover operators and the complicated interactions to mutation, crossover operators are hard to analyze and thus have few theoretical results. Therefore, analyzing crossover not only helps in understanding EAs, but also helps in developing novel techniques for analyzing sophisticated metaheuristic algorithms. In this paper, we derive the General Markov Chain Switching Theorem (GMCST) to facilitate theoretical studies of crossover-enabled EAs. The theorem allows us to analyze the running time of a sophisticated EA from an easy-to-analyze EA. Using this tool, we analyze EAs with several crossover operators on the LeadingOnes and OneMax problems, which are noticeably two well studied problems for mutation-only EAs but with few results for crossover-enabled EAs. We first derive the bounds of running time of the (2+2)-EA with crossover operators; then we study the running time gap between the mutation-only (2:2)-EA and the (2:2)-EA with crossover operators; finally, we develop strategies that apply crossover operators only when necessary, which improve from the mutation-only as well as the crossover-all-the-time (2:2)-EA. The theoretical results are verified by experiments.