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Multiple-gain Estimation for Running Time of Evolutionary Combinatorial Optimization

Min Huang, Pengxiang Chen, Han Huang, Tonli He, Yushan Zhang, Zhifeng Hao

TL;DR

The paper tackles the difficulty of running-time analysis for evolutionary combinatorial optimization by introducing the multiple-gain model, a discrete-time extension of the average gain approach, to bound the expected first hitting time (EFHT). It formalizes a stochastic framework with first hitting times, supermartingales, and the multiple-gain function G(t,k) to derive both worst-case and average-case EFHT bounds via Theorems 1–2 and Corollary 1, using k_low and an auxiliary function h(r). The authors present closed-form EFHT bounds for three canonical instances: (1+1) EA on Onemax, (1+λ) EA on knapsack with favorably correlated weights, and (1+λ) EA on general k-MAX-SAT, along with corresponding k_low values and asymptotic regimes. Experimental results across these problems validate that the theoretical upper bounds align with observed EFHT, confirming the framework as a general and practical tool for running-time analysis in evolutionary combinatorial optimization and suggesting future work for multi-objective extensions.

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+$λ$) EA for the knapsack problem with favorably correlated weights and a closed-form expression of time complexity upper bound in the case of (1+$λ$) 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.

Multiple-gain Estimation for Running Time of Evolutionary Combinatorial Optimization

TL;DR

The paper tackles the difficulty of running-time analysis for evolutionary combinatorial optimization by introducing the multiple-gain model, a discrete-time extension of the average gain approach, to bound the expected first hitting time (EFHT). It formalizes a stochastic framework with first hitting times, supermartingales, and the multiple-gain function G(t,k) to derive both worst-case and average-case EFHT bounds via Theorems 1–2 and Corollary 1, using k_low and an auxiliary function h(r). The authors present closed-form EFHT bounds for three canonical instances: (1+1) EA on Onemax, (1+λ) EA on knapsack with favorably correlated weights, and (1+λ) EA on general k-MAX-SAT, along with corresponding k_low values and asymptotic regimes. Experimental results across these problems validate that the theoretical upper bounds align with observed EFHT, confirming the framework as a general and practical tool for running-time analysis in evolutionary combinatorial optimization and suggesting future work for multi-objective extensions.

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+) EA for the knapsack problem with favorably correlated weights and a closed-form expression of time complexity upper bound in the case of (1+) EA for general -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.
Paper Structure (14 sections, 74 equations, 4 figures, 2 algorithms)

This paper contains 14 sections, 74 equations, 4 figures, 2 algorithms.

Table of Contents

  1. Introduction
  2. Multiple-gain Model
  3. Preliminaries
  4. Average Gain Model and Multiple-Gain Model
  5. Running-time Analysis of EAs Based on Multiple-gain Estimation
  6. Multiple-gain Estimation for EFHT upper bounds of EAs
  7. Running-time Analysis of (1+1) EA for the Onemax Problem
  8. Running-time Analysis of (1+$\lambda$) EA for the Knapsack Problem with Favorably Correlated Weights
  9. Running-time Analysis of (1+$\lambda$) EA for the $k$-MAX-SAT problem
  10. Experiments
  11. Verification of Theorems 3 and 4
  12. Verification of Theorems 5 and 6
  13. Verification of Theorems 7 and 8
  14. Conclusion

Figures (4)

  • Figure 1: Flowchart of the average gain model and multiple-gain model
  • Figure 2: Results of experiment A. (a) Theoretical average-case upper bounds of EFHT and estimation of actual EFHT. (b) Theoretical worst-case upper bounds of EFHT and actual largest FHT. (c) Estimation of $k_{low}$ and theoretical value of $k_{low}$
  • Figure 3: Results of experiment B. (a) Theoretical average-case upper bounds of EFHT and estimation of actual EFHT. (b) Theoretical worst-case upper bounds of EFHT and actual largest FHT. (c) Estimation of $k_{low}$ and theoretical value of $k_{low}$
  • Figure 4: Results of experiment C. (a) Theoretical average-case upper bounds of EFHT and estimation of actual EFHT. (b) Theoretical worst-case upper bounds of EFHT and actual largest FHT. (c) Estimation of $k_{low}$ and theoretical value of $k_{low}$