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.