Rigorous runtime analyses of evolutionary algorithms (EAs) mainly investigate algorithms that use elitist selection methods. Two algorithms commonly studied are Randomized Local Search (RLS) and the (1+1) EA and it is well known that both optimize any linear pseudo-Boolean function on n bits within an expected number of O(n log n) fitness evaluations.In this paper, we analyze variants of these algorithms that use fitness proportional selection.A well-known method in analyzing the local changes in the solutions of RLS is a reduction to the gambler's ruin problem. We extend this method in order to analyze the global changes imposed by the (1+1) EA. By applying this new technique we show that with high probability using fitness proportional selection leads to an exponential optimization time for any linear pseudo-Boolean function with non-zero weights. Even worse, all solutions of the algorithms during an exponential number of fitness evaluations differ with high probability in linearly many bits from the optimal solution.Our theoretical studies are complemented by experimental investigations which confirm the asymptotic results on realistic input sizes.
We show that the natural evolutionary algorithm for the all-pairs shortest path problem is significantly faster with a crossover operator than without. This is the first theoretical analysis proving the usefulness of crossover for a non-artificial problem.
We conduct a rigorous analysis of the (1+1) evolutionary algorithm for the single source shortest path problem proposed by Scharnow, Tinnefeld, and Wegener (The analyses of evolutionary algorithms on sorting and shortest paths problems, 2004, Journal of Mathematical Modelling and Algorithms, 3(4):349–366). We prove that with high probability, the optimization time is O(n2 max{ℓ, log(n)}), where ℓ is the smallest integer such that any vertex can be reached from the source via a shortest path having at most ℓ edges. This bound is tight. For all values of n and ℓ we provide a graph with edge weights such that, with high probability, the optimization time is of order Ω(n2 max{ℓ, log(n)}). To obtain such sharp bounds, we develop a new technique that overcomes the coupon collector behavior of previously used arguments. Also, we exhibit a simple Chernoff type inequality for sums of independent geometrically distributed random variables, and one for sequences of random variables that are not independent, but show a desired behavior independent of the outcomes of the previous random variables. We are optimistic that these tools find further applications in the analysis of evolutionary algorithms.
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