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, Eremeev, and Lehre 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 Dang et al. (in Proceedings of the Genetic and Evolutionary Computation Conference. Association for Computing Machinery, New York, NY, USA, GECCO’21, pp 1133–1141, <a href="https://doi.org/10.1145/3449639.3459398">https://doi.org/10.1145/3449639.3459398</a>, 2021c). 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, denoted Slo<InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(^\alpha _{\varepsilon ,r}\)</EquationSource> </InlineEquation>. The hierarchy is consistent with existing results for the runtime of EAs. The easiest problems are in Slo<InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(^{\alpha }_{\varepsilon }\)</EquationSource> </InlineEquation> for <InlineEquation ID="IEq3"> <EquationSource Format="TEX">\(\alpha =1\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\varepsilon =0\)</EquationSource> </InlineEquation>. As we increase <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(\varepsilon \)</EquationSource> </InlineEquation> and decrease <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(\alpha \)</EquationSource> </InlineEquation>, the function class contains more interesting functions, including instances of hard combinatorial optimisation problems and problems perturbed by static noise. For <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\alpha =O(1/n)\)</EquationSource> </InlineEquation> and <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(\varepsilon =1,\)</EquationSource> </InlineEquation> the problem class contains every problem, including problems closed under permutation (No Free Lunch). Problem classes where local optima sparsity exceed fitness valley density are shown to have exponential black-box complexity. We also 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.

The SLO Hierarchy of Pseudo-Boolean Functions and Runtime of Evolutionary Algorithms

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2026年3月24日

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Duc-Cuong Dang,
Per Kristian Lehre

摘要

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, Eremeev, and Lehre 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 Dang et al. (in Proceedings of the Genetic and Evolutionary Computation Conference. Association for Computing Machinery, New York, NY, USA, GECCO’21, pp 1133–1141, https://doi.org/10.1145/3449639.3459398, 2021c). 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, denoted Slo \(^\alpha _{\varepsilon ,r}\) . The hierarchy is consistent with existing results for the runtime of EAs. The easiest problems are in Slo \(^{\alpha }_{\varepsilon }\) for \(\alpha =1\) and \(\varepsilon =0\) . As we increase \(\varepsilon \) and decrease \(\alpha \) , the function class contains more interesting functions, including instances of hard combinatorial optimisation problems and problems perturbed by static noise. For \(\alpha =O(1/n)\) and \(\varepsilon =1,\) the problem class contains every problem, including problems closed under permutation (No Free Lunch). Problem classes where local optima sparsity exceed fitness valley density are shown to have exponential black-box complexity. We also 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.