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Recently, Dang, Eremeev, and Lehre introduced a classification of pseudo-Boolean problems showing that \u201csparsity\u201d of local optima and the \u201cdensity\u201d 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\u201921, pp 1133\u20131141,\n                    <jats:ext-link xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"10.1145\/3449639.3459398\" ext-link-type=\"doi\">https:\/\/doi.org\/10.1145\/3449639.3459398<\/jats:ext-link>\n                    , 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\n                    <jats:sc>Slo<\/jats:sc>\n                    <jats:inline-formula>\n                      <jats:tex-math>$$^\\alpha _{\\varepsilon ,r}$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . The hierarchy is consistent with existing results for the runtime of EAs. The easiest problems are in\n                    <jats:sc>Slo<\/jats:sc>\n                    <jats:inline-formula>\n                      <jats:tex-math>$$^{\\alpha }_{\\varepsilon }$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    for\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\alpha =1$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\varepsilon =0$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    . As we increase\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\varepsilon $$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    and decrease\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\alpha $$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    , the function class contains more interesting functions, including instances of hard combinatorial optimisation problems and problems perturbed by static noise. For\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\alpha =O(1\/n)$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    and\n                    <jats:inline-formula>\n                      <jats:tex-math>$$\\varepsilon =1,$$<\/jats:tex-math>\n                    <\/jats:inline-formula>\n                    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\n                    <jats:sc>OneMax<\/jats:sc>\n                    with constant probability leads to a problem class that can still be optimised efficiently with appropriately tuned non-elitist EAs.\n                  <\/jats:p>","DOI":"10.1007\/s00453-025-01359-z","type":"journal-article","created":{"date-parts":[[2026,3,24]],"date-time":"2026-03-24T12:37:09Z","timestamp":1774355829000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["The SLO Hierarchy of Pseudo-Boolean Functions and Runtime of Evolutionary Algorithms"],"prefix":"10.1007","volume":"88","author":[{"given":"Duc-Cuong","family":"Dang","sequence":"first","affiliation":[]},{"given":"Per Kristian","family":"Lehre","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2026,3,24]]},"reference":[{"key":"1359_CR1","doi-asserted-by":"publisher","unstructured":"Aita, T., Uchiyama, H., Inaoka, T., et al.: Analysis of a local fitness landscape with a model of the rough Mt. 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