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

2026-05-28

AI Summary

This work investigates the runtime of compact genetic algorithms on the truly multi-valued OneMax function, where each dimension takes values from a set of size $ r $ rather than being binary. To analyze this setting, the authors develop an improved drift theorem and a novel concentration inequality tailored to processes with high self-loop probabilities, enabling a fine-grained characterization of how the probability mass in the frequency matrix concentrates toward the optimal region. These analytical tools yield a significant improvement over the previous best runtime bound, reducing it from $ O(n r^3 \log^2 n \log r) $ to $ O(n r \log^3 n \log^3 r) $. Under the optimal update strength $ K $, this bound approaches the known optimal runtime for the simplified multi-valued case, thereby establishing—for the first time—that the compact genetic algorithm retains near-optimal efficiency even in genuinely multi-valued search spaces.

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.

Problem

• runtime analysis
• compact genetic algorithm
• multi-valued OneMax
• estimation-of-distribution algorithms
• truly multi-valued

Innovation

• compact genetic algorithm
• multi-valued optimization
• runtime analysis
• drift theorems
• concentration inequalities