Recently, the runtime analysis of multi-valued estimation-of-distribution algorithms in the framework of Ben Jedidia et al. (TCS 2024) has made significant advancements. However, almost all existing analyses are limited to multi-valued objective functions that in each dimension only distinguish between two types, also called categories, of values and hence can be treated with similar methods as pseudo-Boolean problems. Only recently, Adak and Witt (GECCO 2025) have presented a first runtime analysis of a multi-valued compact genetic algorithm (cGA) on the multi-valued OneMax function G-OneMax$\colon \{0,\dots,r-1\}^n \to \mathbf{N}$ defined by G-OneMax$(x_1,\dots,x_n)=\sum_{i=1}^n {x}_i$ and truly depending on all $r$ categories. We improve their runtime result from $\textrm{O}\bigl(n r^3 \log^2( n)\log (r)\bigr)$ to $\textrm{O}\bigl(n r \log^3(n)\log^3(r)\bigr)$, both for an optimal choice of the update strength $K$. Our result matches, up to polylogarithmic factors, the existing bound for the simpler $r$-valued OneMax function depending essentially only on two values and analyzed in several previous works. To show the new bound, we use improved drift theorems for processes with high self-loop probabilities and specifically derived concentration inequalities to analyze how probability mass in the multi-valued cGA moves into successively smaller and smaller intervals of the $r$-valued frequency matrix.

翻译：近期，Ben Jedidia等人（TCS 2024）框架下多值分布估计算法的运行时间分析取得了重要进展。然而，现有分析几乎都局限于每个维度仅区分两类（即类别）值的多值目标函数，因此可借助类似伪布尔问题的方法处理。直到最近，Adak与Witt（GECCO 2025）才首次对多值紧凑遗传算法（cGA）在G-OneMax$\colon \{0,\dots,r-1\}^n \to \mathbf{N}$（定义为G-OneMax$(x_1,\dots,x_n)=\sum_{i=1}^n {x}_i$）这一真正依赖全部$r$个类别的多值OneMax函数上进行了运行时间分析。我们将其运行时间结果从$\textrm{O}\bigl(n r^3 \log^2( n)\log (r)\bigr)$改进至$\textrm{O}\bigl(n r \log^3(n)\log^3(r)\bigr)$，两者均基于更新强度$K$的最优选择。该结果在多项式对数因子范围内，匹配了多个前期工作中针对仅依赖两个值的简化$r$值OneMax函数所建立的界限。为证明这一新界，我们运用了针对高自环概率过程的改进漂移定理以及专门推导的浓度不等式，分析多值cGA中概率质量如何逐步向$r$值频率矩阵中逐渐缩小的区间迁移。