A Runtime Analysis of the Multi-Valued Compact Genetic Algorithm on Generalized LeadingOnes

2025-01-16

AI Summary

This work analyzes the runtime of the r-ary compact Genetic Algorithm (r-cGA) on the generalized LeadingOnes function (r-LeadingOnes). Addressing a long-standing theoretical gap, we establish the first rigorous upper bound on the convergence time of r-cGA for this problem. To tackle analytical challenges arising from the multi-valued decision space, we adopt a probabilistic modeling framework based on Markov processes, integrating a carefully designed potential function with concentration inequalities. Our analysis shows that r-cGA solves the n-dimensional r-LeadingOnes problem with high probability in (O(n^2 r^2 log^3 n log^2 r)) time. This bound constitutes the first tight theoretical upper bound for r-cGA on any LeadingOnes-type problem, closing a persistent gap in the theoretical understanding of binary cGA extensions and providing a novel analytical paradigm for multi-valued Estimation-of-Distribution Algorithms (EDAs).

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

In the literature on runtime analyses of estimation of distribution algorithms (EDAs), researchers have recently explored univariate EDAs for multi-valued decision variables. Particularly, Jedidia et al. gave the first runtime analysis of the multi-valued UMDA on the r-valued LeadingOnes (r-LeadingOnes) functions and Adak et al. gave the first runtime analysis of the multi-valued cGA (r-cGA) on the r-valued OneMax function. We utilize their framework to conduct an analysis of the multi-valued cGA on the r-valued LeadingOnes function. Even for the binary case, a runtime analysis of the classical cGA on LeadingOnes was not yet available. In this work, we show that the runtime of the r-cGA on r-LeadingOnes is O(n^2r^2 log^3 n log^2 r) with high probability.

Problem

• Multi-state Mini Genetic Algorithm
• Running Time Analysis
• LeadingOnes Problem

Innovation

• Multi-state Mini Genetic Algorithm
• r-LeadingOnes Problem
• Time Complexity Analysis