Exact computation of the expectation curves for uniform crossover

Authors:
Francisco Chicano;Darrell Whitley;Enrique Alba
Affiliations:
University of Malaga, Malaga, Spain;Colorado State University, Fort Collins, CO, USA;University of Malaga, Malaga, Spain
Venue:
Proceedings of the 14th annual conference on Genetic and evolutionary computation
Year:
2012

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Theoretical Computer Science

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Abstract

Uniform crossover is a popular operator used in genetic algorithms to combine two tentative solutions of a problem represented as binary strings. We use the Walsh decomposition of pseudo-Boolean functions and properties of Krawtchouk matrices to exactly compute the expected value for the fitness of a child generated by uniform crossover from two parent solutions. We prove that this expectation is a polynomial in Á, the probability of selecting the best-parent bit. We provide efficient algorithms to compute this polynomial for ONEMAX and MAX-kSAT problems, but the results also hold for domains such as NK-Landscapes.