Inventory of the 12 007 Low-Dimensional Pseudo-Boolean Landscapes Invariant to Rank, Translation, and Rotation

2026-04-07

AI Summary

This work proposes a robust landscape equivalence relation that simultaneously incorporates invariance under permutation, translation, and rotation, enabling a systematic classification of all pseudo-Boolean optimization functions—including non-injective cases—in dimensions one through three. By combining combinatorial enumeration with exhaustive verification, the study constructs for the first time a complete set of 12,007 invariant landscape classes, substantially fewer than those obtained under permutation invariance alone. The analysis reveals that non-injective functions dominate landscape diversity and elucidates intricate relationships among neutrality, deception, and the performance of hill-climbing algorithms. These findings provide a foundational resource for benchmark design and theoretical investigations in discrete optimization.

Abstract

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.

Problem

• pseudo-Boolean landscapes
• rank invariance
• landscape equivalence
• translation and rotation symmetry
• non-injective functions

Innovation

• rank landscape invariance
• pseudo-Boolean functions
• landscape equivalence
• non-injective functions
• optimization benchmarking