Academic

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

arXiv:2604.05530v1 Announce Type: new 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 p

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Arnaud Liefooghe (LISIC), S\'ebastien Verel (LISIC) · · 1 min read · 4 views
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arXiv:2604.05530v1 Announce Type: new 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.

Executive Summary

The article presents a groundbreaking inventory of 12,007 distinct low-dimensional pseudo-Boolean landscape classes, defined by invariance to rank, translation, and rotation. By extending beyond traditional rank-invariant landscapes, the authors introduce a novel framework that captures neighborhood structures and symmetries, revealing significant reductions in complexity compared to rank-invariant alone. The study exhaustively classifies these landscapes for dimensions 1, 2, and 3, including non-injective functions, which exhibit far greater diversity than injective counterparts. The findings expose intricate relationships between topological properties, algorithmic behaviors, and landscape hardness, offering a robust foundation for benchmark design, pedagogical applications, and the systematic construction of optimization problems with controlled difficulty. This work advances both theoretical understanding and practical applications in randomized optimization and landscape analysis.

Key Points

  • ▸ Introduces a stronger notion of rank landscape invariance that incorporates neighborhood structure and symmetries (translation/rotation), enabling a more nuanced classification of pseudo-Boolean functions.
  • ▸ Provides an exhaustive inventory of 12,007 invariant landscape classes for dimensions 1, 2, and 3, including non-injective functions, revealing far greater complexity than previously recognized.
  • ▸ Highlights emergent properties such as deceptiveness, neutrality, and hill-climbing performance, offering insights into algorithmic behavior and landscape difficulty.

Merits

Theoretical Rigor

The study employs a rigorous mathematical framework to define and classify landscape invariances, extending beyond prior work that focused solely on rank-invariant landscapes.

Comprehensive Classification

The exhaustive inventory for dimensions 1, 2, and 3, including non-injective functions, provides an unprecedented level of detail and completeness in landscape analysis.

Practical Utility

The findings offer a resource for benchmark design and pedagogical applications, enabling the construction of optimization problems with controlled hardness.

Demerits

Scalability Concerns

The classification is limited to low-dimensional landscapes (1, 2, and 3), and the methodology may not scale efficiently to higher dimensions, potentially limiting its applicability.

Assumption of Invariance

The study assumes that landscape invariance under translation and rotation is universally meaningful, which may not hold for all optimization algorithms or problem domains.

Non-Injective Focus

While non-injective functions are included, the study does not fully explore their implications for real-world optimization problems, which may exhibit noise or stochasticity.

Expert Commentary

This article represents a significant advancement in the theory of optimization landscapes by introducing a more nuanced framework for classifying pseudo-Boolean functions. The distinction between rank-invariant and fully invariant landscapes is particularly insightful, as it captures the interplay between problem structure and algorithmic behavior in a way that prior work has overlooked. The exhaustive inventory of 12,007 classes is a tour de force, demonstrating both mathematical rigor and practical utility. Notably, the inclusion of non-injective functions reveals a richness in landscape diversity that was previously underappreciated, challenging the conventional wisdom that injective functions are the primary source of complexity. The study’s focus on topological properties and algorithmic performance also opens new avenues for research, particularly in understanding how landscape features influence optimization outcomes. While the low-dimensional focus is a limitation, the methodology and insights are transferable and could inspire future work in higher dimensions. Overall, this work is a landmark contribution that bridges theory and practice in optimization.

Recommendations

  • ✓ Future research should extend the classification to higher dimensions, exploring whether the patterns observed in low-dimensional landscapes hold in more complex settings.
  • ✓ Investigate the applicability of the invariance framework to other classes of functions, such as continuous or mixed-integer problems, to broaden its impact.
  • ✓ Develop computational tools or software libraries to facilitate the use of the inventory in benchmark design and algorithm selection, making the findings more accessible to practitioners.

Sources

Original: arXiv - cs.AI

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