[Paper Review] Topological Representation of the Transit Sets of k-Point Crossover Operators

Manoj Changat, Prasanth G. Narasimha-Shenoi+5 more|arXiv (Cornell University)|Dec 25, 2017

Advanced Control Systems Optimization33 references3 citations

TL;DR

This paper establishes that k-point crossover operators in genetic algorithms generate transit sets whose convexity structure matches the geodesic convexity of the underlying Hamming graph, resolving an open problem by Mulder. It further shows these transit sets are topes of uniform oriented matroids of VC-dimension k+1, enabling representation via pseudosphere arrangements, with detailed characterization for k=2 using graph-theoretic and topological tools.

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.

Motivation & Objective

• To resolve Mulder's open problem on whether geodesic convexity in graphs is uniquely determined by the interval function.
• To characterize the transit functions of k-point crossover operators and their underlying graph structures.
• To investigate the topological and combinatorial properties of k-point crossover transit sets, particularly for binary strings.
• To determine whether the conjecture by Gitchoff and Wagner—that each transit set (except hypercubes) arises from a unique parent pair—holds true.
• To explore the connection between k-point crossover and oriented matroid theory, especially in terms of VC-dimension and pseudosphere representations.

Proposed method

• Define k-point crossover via interval-based recombination with up to k breakpoints, ensuring parental strings are included in offspring sets.
• Use transit function axioms (T1–T3) to model recombination sets, aligning with Mulder’s framework for convexity and betweenness in graphs.
• Characterize the underlying graphs of k-point crossover transit functions as Hamming graphs or partial cubes.
• Establish that transit sets of k-point crossover on binary strings are topes of uniform oriented matroids with VC-dimension k+1.
• Apply the Topological Representation Theorem for oriented matroids to represent k-point crossover via pseudosphere arrangements.
• Analyze the special case k=2 using graph-theoretic tools, including degree distribution and edge counting in transit set graphs.

Experimental results

Research questions

• RQ1 Does the interval function of a connected graph uniquely determine its geodesic convexity?
• RQ2 Are transit sets of k-point crossover operators on binary strings realizable as topes of uniform oriented matroids?
• RQ3 For k > 1, do k-point crossover transit sets have maximal VC-dimension among all transit functions?
• RQ4 Is the conjecture by Gitchoff and Wagner—that each transit set (except hypercubes) arises from a unique parent pair—true?
• RQ5 Can k-point crossover operators be topologically represented via pseudosphere arrangements, particularly for k=2?

Key findings

• The transit functions of k-point crossover operators (k > 1) generate the same convexity as the geodesic interval function of the underlying Hamming graph, thereby resolving Mulder’s open problem in the negative.
• For k > 1, the transit sets of k-point crossover on binary strings are precisely the topes of a uniform oriented matroid of VC-dimension k+1.
• The Topological Representation Theorem implies that k-point crossover operators can be represented by arrangements of pseudospheres, providing a topological realization of the recombination process.
• For k=2, the transit set graph has 2t vertices of degree 3 and t²−3t vertices of degree 4, derived from a system of linear equations based on the handshaking lemma.
• The transit sets of k-point crossover operators are shown to have maximal VC-dimension among all transit functions, confirming their combinatorial complexity.
• The results extend directly to all Hamming graphs, as the structure of k-point crossover depends only on Hamming distance and string position differences between parents.