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

TL;DR

This paper provides an exhaustive classification of 12,007 low-dimensional pseudo-Boolean landscape classes invariant to rank, translation, and rotation, aiding benchmark design and understanding landscape complexity.

Contribution

It introduces a comprehensive inventory of invariant landscape classes for pseudo-Boolean functions up to dimension 3, highlighting the diversity and properties of these classes.

Findings

Non-injective functions produce more invariant classes than injective ones.

Complex landscape properties influence algorithm performance and deceptiveness.

The inventory aids in benchmark design and understanding landscape difficulty.

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.