Enumerating maximal consistent closed sets in closure systems

Lhouari Nourine; Simon Vilmin

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arXiv · 2102.04245

Enumerating maximal consistent closed sets in closure systems

Lhouari Nourine·Simon Vilmin

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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.

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Lhouari Nourine, Simon Vilmin. 2021-02-05. Enumerating maximal consistent closed sets in closure systems. https://arxiv.org/abs/2102.04245

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