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

TL;DR

This paper characterizes the optimal implicational bases of convex geometries using quasi-closed hypergraphs, providing polynomial-time optimization methods for certain classes and unifying previous tractability results.

Contribution

It introduces quasi-closed hypergraphs to characterize optimal bases and identifies conditions under which optimization is polynomial-time solvable.

Findings

Polynomial-time optimization for convex geometries with disjoint quasi-closed hypergraph edges.

Characterization of optimal bases via quasi-closed hypergraphs.

Unification of tractability results for double-shelling, acyclic, affine, and acceptant convex geometries.

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.