The SLO Hierarchy of pseudo-Boolean Functions and Runtime of Evolutionary Algorithms
Pages 1551 - 1559
Abstract
While some common fitness landscape characteristics are critical when determining the runtime of evolutionary algorithms (EAs), the relationship between fitness landscape structure and the runtime of EAs is poorly understood. Recently, Dang et al. (2021) introduced a classification of pseudo-Boolean problems showing that "sparsity" of local optima and the "density" of fitness valleys can be crucial characteristics when determining the runtime of EAs. However, their approach could only classify some classes of pseudo-Boolean functions and thus defined an incomplete hierarchy.
We generalise the previous work to a complete hierarchy for all pseudo-Boolean functions. The hierarchy is consistent with existing results for the runtime of EAs. The hardest part of the hierarchy consists of problems satisfying the No Free Lunch theorem. The easiest part contains well-known theoretical benchmark problems, easy for EAs. The intermediary parts contain instances of NP-hard problems. Problem classes where local optima sparsity exceed fitness valley density are shown to have exponential black-box complexity. We study how random perturbations of a function can change its classification. E.g, randomly perturbing search points in OneMax with constant probability leads to a problem class that can still be optimised efficiently with appropriately tuned non-elitist EAs.
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Published: 14 July 2024
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