Lower Bounds for Non-elitist Evolutionary Algorithms via Negative Multiplicative Drift

摘要

A decent number of lower bounds for non-elitist population-based evolutionary algorithms has been shown by now. Most of them are technically demanding due to the (hard to avoid) use of negative drift theorems - general results which translate an expected progress away from the target into a high hitting time. We propose a simple negative drift theorem for multiplicative drift scenarios and show that it can simplify existing analyses. We discuss in more detail Lehre's (PPSN 2010) negative drift in populations method, one of the most general tools to prove lower bounds on the runtime of non-elitist mutation-based evolutionary algorithms for discrete search spaces. Together with other arguments, we obtain an alternative and simpler proof, which also strengthens and simplifies this method. In particular, now only three of the five technical conditions of the previous result have to be verified. The lower bounds we obtain are explicit instead of only asymptotic. This allows to compute concrete lower bounds for concrete algorithms, but also enables us to show that super-polynomial runtimes appear already when the reproduction rate is only a (1-ω(n~(-1/2))) factor below the threshold. As one particular result, we apply this method and a novel domination argument to show an exponential lower bound for the runtime of the mutation-only simple GA on OneMax for arbitrary population size.