Work

Translating Between the Representations of an Acyclic Convex Geometry of Bounded Degree

Year: 2026

Type: article

Abstract: We consider the problem of translating between irreducible closed sets and implicational bases in closure systems. To date, the complexity status of this problem is widely open, and it is further known to generalize the notorious hypergraph dualization problem, even in the case of acyclic convex geometries, i.e., closure systems admitting an acyclic implicational base. This paper studies this case with a focus on the degree, which corresponds to the maximal number of implications in which an element occurs. We show that the problem is tractable for bounded values of this parameter, even when relaxed to the notions of premise- and conclusion-degree. Our algorithms rely on structural properties of acyclic convex geometries and involve various techniques from algorithmic enumeration such as solution graph traversal, saturation techniques, and a sequential approach leveraging from acyclicity. They are shown to perform in incremental-polynomial time. Finally, we complete these results by showing that our running times cannot be improved to polynomial delay using the standard framework of flashlight search.

Source: The Electronic Journal of Combinatorics

DOI: 10.37236/14966

Authors: Oscar Defrain, Arthur Ohana, Simon Vilmin

Institutions: Centre National de la Recherche Scientifique, Aix-Marseille Université, Laboratoire d’Informatique et Systèmes, Université Grenoble Alpes

Language: English

Topic: Advanced Graph Theory Research

Subfield: Computational Theory and Mathematics

Field: Computer Science

Domain: Physical Sciences

Open Access status: diamond

Funder: Agence Nationale de la Recherche

Award: Parameterized aspects of Dualization