Complexity Results for Implication Bases of Convex Geometries

TL;DR

This paper investigates the computational complexity of representing convex geometries with implication bases, proving NP-hardness for optimization and co-NP-completeness for recognition problems, highlighting inherent computational challenges.

Contribution

It establishes the NP-hardness of optimizing implication bases and co-NP-completeness of recognizing convex geometries, advancing understanding of their computational complexity.

Findings

Implication basis optimization is NP-hard.

Deciding if an implication basis defines a convex geometry is co-NP-complete.

Provides complexity bounds for key problems in convex geometry representations.

Abstract

A convex geometry is finite zero-closed closure system that satisfies the anti-exchange property. Complexity results are given for two open problems related to representations of convex geometries using implication bases. In particular, the problem of optimizing an implication basis for a convex geometry is shown to be NP-hard by establishing a reduction from the minimum cardinality generator problem for general closure systems. Furthermore, even the problem of deciding whether an implication basis defines a convex geometry is shown to be co-NP-complete by a reduction from the Boolean tautology problem.