Improved Runtime Bound for the $(μ+ 1)$ EA on BinVal

2026-06-11

AI Summary

This study investigates the upper bound on the expected running time of the $(\mu+1)$ evolutionary algorithm (EA) on the BinVal function. For the regime where $\mu = o(n/\log n)$, the authors significantly improve the best-known upper bound from $O(\mu^5 n \log(n/\mu^4))$ to $O(\mu \log \mu \cdot n \log n)$ through rigorous probabilistic analysis and runtime complexity theory. The result holds for various mutation operators, including standard bit mutation, and demonstrates that the optimization efficiency of the $(\mu+1)$ EA on BinVal differs from that on OneMax by at most a logarithmic factor of $O(\log \mu \cdot \log n)$. This indicates that the two problems exhibit nearly identical performance under this algorithmic framework.

Abstract

We study the $(μ+1)$ EA on the Binary Value function BinVal. We show that it needs at most $O(μ\log μ\cdot n \log n)$ function evaluations to find the optimum when $μ= o(n/\log n)$. This substantially improves upon the recent upper bound of $O(μ^5 n \log(n/μ^4))$ by Krejca, Neumann and Witt. Our results hold for several mutation operators including standard bit mutation. In particular, our bound implies that the $(μ+1)$ EA is at most a factor $O(\log μ\cdot \log n)$ slower on BinVal than on OneMax.

Problem

• evolutionary algorithm
• runtime analysis
• BinVal
• upper bound
• population size

Innovation

• evolutionary algorithms
• runtime analysis
• BinVal
• upper bound
• standard bit mutation