Estimate Hitting Time by Hitting Probability for Elitist Evolutionary Algorithms

TL;DR

This paper introduces a novel drift analysis method based on hitting probability to efficiently estimate hitting times of elitist evolutionary algorithms, simplifying analysis and enabling performance comparison.

Contribution

It proposes a new approach that transforms hitting time estimation into hitting probability estimation, with explicit formulas and path-based analysis for multimodal landscapes.

Findings

The method can estimate both lower and upper bounds of hitting time.

It allows comparison of algorithm performance in terms of hitting time.

Applied to knapsack algorithms, it shows no consistent superiority of constraint handling techniques.

Abstract

Drift analysis is a powerful tool for analyzing the time complexity of evolutionary algorithms. However, it requires manual construction of drift functions to bound hitting time for each specific algorithm and problem. To address this limitation, general linear drift functions were introduced for elitist evolutionary algorithms. But calculating linear bound coefficients effectively remains a problem. This paper proposes a new method called drift analysis of hitting probability to compute these coefficients. Each coefficient is interpreted as a bound on the hitting probability of a fitness level, transforming the task of estimating hitting time into estimating hitting probability. A novel drift analysis method is then developed to estimate hitting probability, where paths are introduced to handle multimodal fitness landscapes. Explicit expressions are constructed to compute hitting probability, significantly simplifying the estimation process. One advantage of the proposed method is its ability to estimate both the lower and upper bounds of hitting time and to compare the performance of two algorithms in terms of hitting time. To demonstrate this application, two algorithms for the knapsack problem, each incorporating feasibility rules and greedy repair respectively, are compared. The analysis indicates that neither constraint handling technique consistently outperforms the other.