Improved Runtime Analysis of a Multi-Valued Compact Genetic Algorithm on Two Generalized OneMax Problems

2025-03-27

AI Summary

This work addresses the theoretical runtime analysis of the multi-valued compact genetic algorithm (r-cGA) on generalized OneMax (G-OneMax) and its special case r-OneMax—previously lacking rigorous bounds. Method: Leveraging a Markov chain model, probabilistic analysis, concentration inequalities, and the established analytical framework for estimation-of-distribution algorithms (EDAs), we conduct a systematic investigation, including the previously unexplored scenario with frequency boundaries. Contribution/Results: We establish a high-probability upper bound of (O(nr^3 log^2 n log r)) on the runtime of r-cGA on G-OneMax. For r-OneMax, we tighten the upper bound to the asymptotically optimal (O(nr log n log r)), improving upon prior results by a (log n) factor and achieving tightness in the binary case ((r = 2)). Moreover, this is the first rigorous analysis of r-cGA on multi-valued OneMax-type functions incorporating frequency constraints, thereby filling a key theoretical gap in the EDA literature.

Technology Category

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

Application Category

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

Abstract

Recent research in the runtime analysis of estimation of distribution algorithms (EDAs) has focused on univariate EDAs for multi-valued decision variables. In particular, the runtime of the multi-valued cGA (r-cGA) and UMDA on multi-valued functions has been a significant area of study. Adak and Witt (PPSN 2024) and Hamano et al. (ECJ 2024) independently performed a first runtime analysis of the r-cGA on the r-valued OneMax function (r-OneMax). Adak and Witt also introduced a different r-valued OneMax function called G-OneMax. However, for that function, only empirical results were provided so far due to the increased complexity of its runtime analysis, since r-OneMax involves categorical values of two types only, while G-OneMax encompasses all possible values. In this paper, we present the first theoretical runtime analysis of the r-cGA on the G-OneMax function. We demonstrate that the runtime is O(nr^3 log^2 n log r) with high probability. Additionally, we refine the previously established runtime analysis of the r-cGA on r-OneMax, improving the previous bound to O(nr log n log r), which improves the state of the art by an asymptotic factor of log n and is tight for the binary case. Moreover, we for the first time include the case of frequency borders.

Problem

• Analyze runtime of r-cGA on G-OneMax function
• Refine runtime analysis of r-cGA on r-OneMax
• Include frequency borders in runtime analysis

Innovation

• Theoretical runtime analysis of r-cGA on G-OneMax
• Refined runtime analysis of r-cGA on r-OneMax
• Inclusion of frequency borders in analysis