Enumerating Maximal Consistent Closed Sets in Closure Systems

Lhouari Nourine (1, 2) , Simon Vilmin (1, 2)

1 UCA - Université Clermont Auvergne
2 LIMOS - Laboratoire d'Informatique, de Modélisation et d'Optimisation des Systèmes

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 P = NP, even for lower bounded lattices. We give an incremental-polynomial time algorithm to solve MCCEnum for closure systems with constant Carathéodory 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 is solved by a polynomial delay algorithm [22, 25].

Mots clés: Closure systems; implicational base; inconsistency relation; enumeration algorithm

Domaines: Informatique [cs]

Lhouari Nourine, Simon Vilmin. Enumerating Maximal Consistent Closed Sets in Closure Systems. Formal Concept Analysis. ICFCA 2021, 12733, pp.57-73, 2021, Lecture Notes in Computer Science, ⟨10.1007/978-3-030-77867-5_4⟩. ⟨hal-03354969⟩