Transit sets of two-point crossover

Abstract

Genetic Algorithms typically invoke crossover operators to two parents. The transit set R k ( x, y ) comprises all offsprings of this form. It forms the tope set of an uniform oriented matroid with Vapnik-Chervonenkis dimension k + 1 . The Topological Representation Theorem for oriented matroids thus implies a representation in terms of pseudosphere arrangements. This makes it possible to study 2 -point crossover in detail and to characterize the partial cubes defined by the transit sets of two-point crossover.