Multiple-gain Estimation for Running Time of Evolutionary Combinatorial Optimization

TL;DR

This paper introduces a multiple-gain model to more accurately estimate the running time of evolutionary algorithms in combinatorial optimization, providing tighter bounds and a unified analysis approach.

Contribution

It proposes an improved multiple-gain model that generalizes existing methods, yielding new bounds and closed-form expressions for various combinatorial problems.

Findings

Briefer proof for (1+1) EA time complexity on Onemax

Tighter upper bounds for (1+λ) EA on knapsack problem

Closed-form upper bound for (1+λ) EA on k-MAX-SAT

Abstract

The running-time analysis of evolutionary combinatorial optimization is a fundamental topic in evolutionary computation. Its current research mainly focuses on specific algorithms for simplified problems due to the challenge posed by fluctuating fitness values. This paper proposes a multiple-gain model to estimate the fitness trend of population during iterations. The proposed model is an improved version of the average gain model, which is the approach to estimate the running time of evolutionary algorithms for numerical optimization. The improvement yields novel results of evolutionary combinatorial optimization, including a briefer proof for the time complexity upper bound in the case of (1+1) EA for the Onemax problem, two tighter time complexity upper bounds than the known results in the case of (1+$\lambda$) EA for the knapsack problem with favorably correlated weights and a closed-form expression of time complexity upper bound in the case of (1+$\lambda$) EA for general $k$-MAX-SAT problems. The results indicate that the practical running time aligns with the theoretical results, verifying that the multiple-gain model is more general for running-time analysis of evolutionary combinatorial optimization than state-of-the-art methods.