Transit sets of -point crossover operators

Abstract

k-point crossover operators and their recombination sets are studied from different perspectives. We show that transit functions of k-point crossover generate, for all k > 1, the same convexity as the interval function of the underlying graph. This settles in the negative an open problem by Mulder about whether the geodesic convexity of a connected graph G is uniquely determined by its interval function I. The conjecture of Gitchoff and Wagner that for each transit set R k (x, y) distinct from a hypercube there is a unique pair of parents from which it is generated is settled affirmatively. Along the way we characterize transit functions whose underlying graphs are Hamming graphs, and those with underlying partial cube graphs. For general values of k it is shown that the transit sets of k-point crossover operators are the subsets with maximal Vapnik-Chervonenkis dimension. Moreover, the transit sets of k-point crossover on binary strings form topes of uniform oriented matroid of VC-dimension k + 1. The Topological Representation Theorem for oriented matroids therefore implies that k-point crossover operators can be represented by pseudosphere arrangements. This provides the tools necessary to study the special case k = 2 in detail.

A Unifying View on Recombination Spaces and Abstract Convex Evolutionary Search

“…topologically equivalent) to mutation spaces on graphs under certain conditions [5]. Figure 2 also illustrates via hyperedges of R Ω hypergraph the following coincidence: offspring produced by the uniform recombination P-structure R Ω and Hamming segments are equivalent [11], and, independently, [10,13] show it for Hamming segments, convex hulls and traditional schema [14]. For instance, the schema 0** corresponding to the hyperedge {000, 001, 010, 011} is obtained by R Ω (010, 001), [010; 001] d H or co({010, 001}).…”

“…This naturally leads to formalise the search space as a graph, with the structure given by the adjacency matrix. To model recombination, EL uses an algebraic notion of interval (Definition 6) that captures general recombination structures, not just genetic crossovers [11,12].…”

“…is a recombination P-structure [11]. For instance, consider x := 100 and y := 001 in H 3 2 , then we have offspring: R Ω (100, 001) = {100, 001, 000, 101}.…”

“…Besides, recombination P-structures can be used to define closures that are equivalent to convex hulls [11,12]. One may attempt to define one by taking the union over all pairs of points for a given set, and collect their intervals given by a P-structure (Fig.…”

Axiomatic characterization of transit functions of weak hierarchies

“…Several classes of interesting transit function associated with connected graphs have been studied from this perspective [15], among them the interval function [1,24,26], induced path function [12,13,23], the all paths function [11], the pre-fiber transit function [25], P 3 -transit function [18]. Transit function also arose as models of recombination operators in genetics and evolutionary algorithms [16,22,27], where again, encapsulate the idea that offsprings are genetically "in-between" their parents.…”