Runtime Analysis of a Compact Genetic Algorithm on a Truly Multi-valued OneMax Function

• URL: http://arxiv.org/abs/2605.29477v2
• Date: Mon, 01 Jun 2026 13:55:11 GMT
• Title: Runtime Analysis of a Compact Genetic Algorithm on a Truly Multi-valued OneMax Function
• Abstract summary: 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.<n>We improve their runtime result from $textrmObigl(n r3 log2, n)log (r)bigr)$ to $textrmObigl(n rlog3(n)log3(r)bigr)$ for an optimal choice of the update strength $K$.
• Score: 5.478764356647438
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: 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.

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