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

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.

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

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.

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

A class of metaheuristic techniques called estimation-of-distribution algorithms (EDAs) are employed in optimization as more sophisticated substitutes for traditional strategies like evolutionary algorithms. EDAs generally drive the search for the optimum by creating explicit probabilistic models of potential candidate solutions through repeated sampling and selection from the underlying search space. Most theoretical research on EDAs has focused on pseudo-Boolean optimization. Jedidia et al. (GECCO 2023) proposed the first EDAs for optimizing problems involving multi-valued decision variables. By building a framework, they have analyzed the runtime of a multi-valued UMDA on the r-valued LeadingOnes function. Using their framework, here we focus on the multi-valued compact genetic algorithm (r-cGA) and provide a first runtime analysis of a generalized OneMax function. To prove our results, we investigate the effect of genetic drift and progress of the probabilistic model towards the optimum. After finding the right algorithm parameters, we prove that the r-cGA solves this r-valued OneMax problem efficiently. We show that with high probability, the runtime bound is O(r2 n log2 r log3 n). At the end of experiments, we state one conjecture related to the expected runtime of another variant of multi-valued OneMax function.