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.