Running-time Analysis of ($\mu+\lambda$) Evolutionary Combinatorial Optimization Based on Multiple-gain Estimation

The running-time analysis of evolutionary combinatorial optimization is a fundamental topic in evolutionary computation. However, theoretical results regarding the $(\mu+\lambda)$ evolutionary algorithm (EA) for combinatorial optimization problems remain relatively scarce compared to those for simple pseudo-Boolean problems. This paper proposes a multiple-gain model to analyze the running time of EAs for combinatorial optimization problems. The proposed model is an improved version of the average gain model, which is a fitness-difference drift approach under the sigma-algebra condition to estimate the running time of evolutionary numerical optimization. The improvement yields a framework for estimating the expected first hitting time of a stochastic process in both average-case and worst-case scenarios. It also introduces novel running-time results of evolutionary combinatorial optimization, including two tighter time complexity upper bounds than the known results in the case of ($\mu+\lambda$) EA for the knapsack problem with favorably correlated weights, a closed-form expression of time complexity upper bound in the case of ($\mu+\lambda$) EA for general $k$-MAX-SAT problems and a tighter time complexity upper bounds than the known results in the case of ($\mu+\lambda$) EA for the traveling salesperson problem. Experimental results indicate that the practical running time aligns with the theoretical results, verifying that the multiple-gain model is an effective tool for running-time analysis of ($\mu+\lambda$) EA for combinatorial optimization problems.