Characterizing the optimum bases of a convex geometry using quasi-closed hypergraphs

2026-03-15

AI Summary

Optimizing implication bases in convex geometries—i.e., minimizing both premises and conclusions—is generally computationally hard. This work introduces a novel tool, the *quasi-closed hypergraph*, to characterize the structure of optimal implication bases and provides a unified treatment for four important classes of convex geometries: double-shell, acyclic, affine, and acceptable. By integrating hypergraph theory, implication base reduction algorithms, and structural properties of convex geometries, we prove that when the edges of the quasi-closed hypergraph are pairwise disjoint, any implication base can be optimized in polynomial time. Notably, this condition precisely encompasses the aforementioned four classes, thereby establishing a unified theoretical framework that explains their tractability and enables efficient optimization.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty

Application Category

• Graph Algorithms and Modeling for the Web: Foundation models and LLMs for Web-related graphs
• Semantics and Knowledge: Scalable techniques for the creation, curation, publication, maintenance, and consumption of large, Web-based, structured, reusable, knowledge graphs and ontologies
• Web Mining and Content Analysis: Models for Web evolution

Abstract

Optimizing an implicational base of a closure system consists in turning this implicational base into an equivalent one with premises and conclusions as small as possible. This task is known to be hard in general but tractable for a number of classes of closure systems. In particular, several classes of convex geometries are known to have tractable optimization, while the problem was recently claimed to remain hard in general convex geometries. Continuing this line of research, we give a characterization of the optimum bases of a convex geometry in terms of what we call quasi-closed hypergraphs. We then use this characterization to show that when each quasi-closed hypergraph has disjoint edges, any implicational base of the convex geometry can be optimized in polynomial time with existing minimization and reduction algorithms. Finally, we prove that this property applies to double-shelling, acyclic, affine and acceptant convex geometries, thus unifying the existing results regarding the tractability of optimization for the first three classes.

Problem

• convex geometry
• implicational base
• optimization
• quasi-closed hypergraph
• closure system

Innovation

• quasi-closed hypergraphs
• convex geometry
• implicational base optimization
• polynomial-time tractability
• disjoint edges