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.

翻译：近年来，分布估计算法（EDAs）的运行时分析研究主要集中于处理多值决策变量的单变量EDAs。特别是，多值紧凑遗传算法（r-cGA）和UMDA在多值函数上的运行时性能已成为重要研究领域。Adak与Witt（PPSN 2024）以及Hamano等人（ECJ 2024）分别独立完成了r-cGA在r值OneMax函数（r-OneMax）上的首次运行时分析。Adak与Witt还提出了另一种称为G-OneMax的r值OneMax函数。然而，由于该函数的运行时分析复杂度更高——r-OneMax仅涉及两种类型的分类值，而G-OneMax涵盖所有可能取值——目前仅提供了实证结果。本文首次对r-cGA在G-OneMax函数上进行了理论运行时分析，证明其运行时以高概率为O(nr^3 log^2 n log r)。此外，我们改进了先前建立的r-cGA在r-OneMax上的运行时分析，将原有界限提升至O(nr log n log r)，该结果通过渐进因子log n改进了现有最优结果，且在二元情况下达到紧界。值得注意的是，我们首次将频率边界情况纳入分析框架。