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Enumerating the Irreducible Closed Sets of an Acyclic Implicational Base of Bounded Degree

Authors Oscar Defrain , Arthur Ohana , Simon Vilmin

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ACM Subject Classification
  • Mathematics of computing → Enumeration
Keywords
  • Algorithmic enumeration
  • closure systems
  • acyclic convex geometries
  • solution graph traversal
  • flashlight search
  • extension
  • hypergraph dualization

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Abstract

We consider the problem of enumerating the irreducible closed sets of a closure system given by an implicational base. 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 a sequential approach leveraging from acyclicity, combined with the solution graph traversal technique for the case of premise-degree. They are shown to perform in incremental-polynomial time. These results are complemented in the long version of this document by showing that the dual problem of constructing the implicational base can be solved in polynomial time. Finally, we argue that our running times cannot be improved to polynomial delay using the standard framework of flashlight search.

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Oscar Defrain, Arthur Ohana, and Simon Vilmin. Enumerating the Irreducible Closed Sets of an Acyclic Implicational Base of Bounded Degree. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 24:1-24:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.24

Author Details

Oscar Defrain
  • Aix-Marseille Université, CNRS, LIS, Marseille, France
Arthur Ohana
  • Aix-Marseille Université, CNRS, LIS, Marseille, France
  • Université Grenoble Alpes, Grenoble, France
Simon Vilmin
  • Aix-Marseille Université, CNRS, LIS, Marseille, France

Funding

The authors have been supported by the ANR project PARADUAL (ANR-24-CE48-0610-01).

Acknowledgements

The authors are grateful to anonymous reviewers for their suggestions to improve this extended abstract as well as the long version.

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