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Evolutionary algorithms (EAs), simulating the evolution process of naturalspecies, are used to solve optimization problems. Crossover (also calledrecombination), originated from simulating the chromosome exchange phenomena inzoogamy reproduction, is widely employed in EAs to generate offspringsolutions, of which the effectiveness has been examined empirically inapplications. However, due to the irregularity of crossover operators and thecomplicated interactions to mutation, crossover operators are hard to analyzeand thus have few theoretical results. Therefore, analyzing crossover not onlyhelps in understanding EAs, but also helps in developing novel techniques foranalyzing 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 allowsus to analyze the running time of a sophisticated EA from an easy-to-analyzeEA. Using this tool, we analyze EAs with several crossover operators on theLeadingOnes and OneMax problems, which are noticeably two well studied problemsfor mutation-only EAs but with few results for crossover-enabled EAs. We firstderive 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 applycrossover operators only when necessary, which improve from the mutation-onlyas well as the crossover-all-the-time (2:2)-EA. The theoretical results areverified by experiments.

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