Estimate Hitting Time by Hitting Probability for Elitist Evolutionary Algorithms

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.

Fast Estimations of Hitting Time of Elitist Evolutionary Algorithms from Fitness Levels

The fitness level method is an easy-to-use tool for estimating the hitting time of elitist EAs. Recently, general linear lower and upper bounds from fitness levels have been constructed. However, the construction of these bounds requires recursive computation, which makes them difficult to use in practice. We address this shortcoming with a new directed graph (digraph) method that does not require recursive computation and significantly simplifies the calculation of coefficients in linear bounds. In this method, an EA is modeled as a Markov chain on a digraph. Lower and upper bounds are directly calculated using conditional transition probabilities on the digraph. This digraph method provides straightforward and explicit expressions of lower and upper time bound for elitist EAs. In particular, it can be used to derive tight lower bound on both fitness landscapes without and with shortcuts. This is demonstrated through four examples: the (1+1) EA on OneMax, FullyDeceptive, TwoMax1 and Deceptive. Our work extends the fitness level method from addressing simple fitness functions without shortcuts to more realistic functions with shortcuts.