Parent Selection Mechanisms in Elitist Crossover-Based Algorithms

TL;DR

This paper introduces a parent selection strategy in genetic algorithms that enhances diversity and significantly improves the expected time to solve Jump$_k$ problems, advancing theoretical understanding of crossover's role.

Contribution

It proposes a novel parent selection method prioritizing distant parents, along with a new diversity metric, and provides a theoretical analysis demonstrating improved performance.

Findings

The new strategy solves Jump$_k$ in $O(k4^kn ext{log}(n))$ expected time.

The bound is significantly better than previous bounds for similar GAs without diversity mechanisms.

Crossover is shown to be effective in maintaining diversity throughout the run.

Abstract

Parent selection methods are widely used in evolutionary computation to accelerate the optimization process, yet their theoretical benefits are still poorly understood. In this paper, we address this gap by proposing a parent selection strategy for the $(\mu+1)$ genetic algorithm (GA) that prioritizes the selection of maximally distant parents for crossover. We show that, with an appropriately chosen population size, the resulting algorithm solves the Jump$_k$ problem in $O(k4^kn\log(n))$ expected time. This bound is significantly smaller than the best known bound of $O(n\mu\log(\mu)+n\log(n)+n^{k-1})$ for any $(\mu+1)$~GA using no explicit diversity-preserving mechanism and a constant crossover probability. To establish this result, we introduce a novel diversity metric that captures both the maximum distance between pairs of individuals in the population and the number of pairs achieving this distance. The main novelty of our analysis is that it relies on crossover as a mechanism for creating and maintaining diversity throughout the run, rather than using crossover only in the final step to combine already diversified individuals. The insights provided by our analysis contribute to a deeper theoretical understanding of the role of crossover in the population dynamics of genetic algorithms.