Enumerating maximal consistent closed sets in closure systems

TL;DR

This paper investigates the computational complexity of enumerating maximal consistent closed sets in closure systems, establishing hardness results and providing efficient algorithms under specific conditions.

Contribution

It proves MCCEnum is not output-polynomial unless P=NP, and offers incremental-polynomial and quasipolynomial algorithms for special classes of closure systems.

Findings

MCCEnum cannot be solved in output-polynomial time unless P=NP.

An incremental-polynomial time algorithm exists for closure systems with constant Carathéodory number.

MCCEnum can be solved in output-quasipolynomial time in biatomic atomistic closure systems with independence conditions.

Abstract

Given an implicational base, a well-known representation for a closure system, an inconsistency binary relation over a finite set, we are interested in the problem of enumerating all maximal consistent closed sets (denoted by MCCEnum for short). We show that MCCEnum cannot be solved in output-polynomial time unless $\textsf{P} = \textsf{NP}$, even for lower bounded lattices. We give an incremental-polynomial time algorithm to solve MCCEnum for closure systems with constant Carath\'eodory number. Finally we prove that in biatomic atomistic closure systems MCCEnum can be solved in output-quasipolynomial time if minimal generators obey an independence condition, which holds in atomistic modular lattices. For closure systems closed under union (i.e., distributive), MCCEnum has been previously solved by a polynomial delay algorithm.