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This paper exhaustively enumerates all rank, translation, and rotation invariant landscapes for pseudo-Boolean functions up to 3 dimensions, identifying 12,007 distinct classes. By considering a stronger notion of landscape invariance that incorporates neighborhood structure and symmetries, the authors significantly reduce the number of landscapes compared to considering rank-invariance alone. The analysis reveals a surprising prevalence of non-injective functions in generating invariant landscape classes and uncovers complex relationships between landscape topology and algorithm performance.
Forget individual functions – the symmetries of low-dimensional landscapes reveal only 12,007 fundamentally different optimization problems for rank-based algorithms.
Many randomized optimization algorithms are rank-invariant, relying solely on the relative ordering of solutions rather than absolute fitness values. We introduce a stronger notion of rank landscape invariance: two problems are equivalent if their ranking, but also their neighborhood structure and symmetries (translation and rotation), induce identical landscapes. This motivates the study of rank landscapes rather than individual functions. While prior work analyzed the rankings of injective function classes in isolation, we provide an exhaustive inventory of the invariant landscape classes for pseudo-Boolean functions of dimensions 1, 2, and 3, including non-injective cases. Our analysis reveals 12,007 classes in total, a significant reduction compared to rank-invariance alone. We find that non-injective functions yield far more invariant landscape classes than injective ones. In addition, complex combinations of topological landscape properties and algorithm behaviors emerge, particularly regarding deceptiveness, neutrality, and the performance of hill-climbing strategies. The inventory serves as a resource for pedagogical purposes and benchmark design, offering a foundation for constructing larger problems with controlled hardness and advancing our understanding of landscape difficulty and algorithm performance.
