Parent Selection Mechanisms in Elitist Crossover-Based Algorithms

2026-04-05

AI Summary

This work addresses the limited theoretical understanding of how parent selection strategies accelerate optimization in evolutionary computation. Within the (μ+1) genetic algorithm framework, the authors introduce a parent selection mechanism that pairs individuals at maximum Hamming distance with probability Ω(1), alongside a novel diversity metric. This approach elucidates the critical role of crossover in sustaining population diversity throughout the entire evolutionary process, moving beyond the conventional focus on final-generation diversity alone. When applied to the Jump_k problem, the proposed method achieves an expected runtime of O(k 4^k n log n), substantially outperforming existing algorithms of the same class that lack explicit diversity-preserving mechanisms.

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 incorporating different parent selection strategies into the $(μ+1)$ genetic algorithm (GA). We show that, with an appropriately chosen population size and a parent selection strategy that selects a pair of maximally distant parents with probability $Ω(1)$ for crossover, 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 $O(nμ\log(μ)+n\log(n)+n^{k-1})$ for any $(μ+1)$~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 crucial point 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, as it has been done in many previous works. The insights provided by our analysis contribute to a deeper theoretical understanding of the role of crossover in the population dynamics of genetic algorithms.

Problem

• parent selection
• crossover
• genetic algorithms
• population diversity
• theoretical analysis

Innovation

• parent selection
• crossover-based diversity
• genetic algorithms
• Jump_k problem
• population diversity metric