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On the Enumeration of Signatures of XOR-CNF’s

Authors Nadia Creignou , Oscar Defrain , Frédéric Olive , Simon Vilmin

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ACM Subject Classification
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Graph enumeration
Keywords
  • Algorithmic enumeration
  • XOR-CNF
  • signatures
  • maximal bipartite subgraphs enumeration
  • extension
  • proximity search

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Abstract

Given a CNF formula φ with clauses C_1, … , C_m over a set of variables V, a truth assignment 𝐚: V → {0, 1} generates a binary sequence σ_φ(𝐚) = (C_1(𝐚), …, C_m(𝐚)), called a signature of φ, where C_i(𝐚) = 1 if clause C_i evaluates to 1 under assignment 𝐚, and C_i(𝐚) = 0 otherwise. Signatures and their associated generation problems have given rise to new yet promising research questions in algorithmic enumeration. In a recent paper, Bérczi et al. interestingly proved that generating signatures of a CNF is tractable despite the fact that verifying a solution is hard. They also showed the hardness of finding maximal signatures of an arbitrary CNF due to the intractability of satisfiability in general. Their contribution leaves open the problem of efficiently generating maximal signatures for tractable classes of CNFs, i.e., those for which satisfiability can be solved in polynomial time. Stepping into that direction, we completely characterize the complexity of generating all, minimal, and maximal signatures for XOR-CNF’s.

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Nadia Creignou, Oscar Defrain, Frédéric Olive, and Simon Vilmin. On the Enumeration of Signatures of XOR-CNF’s. In 19th International Symposium on Algorithms and Data Structures (WADS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 349, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.WADS.2025.19

Author Details

Nadia Creignou
  • Aix-Marseille Université, CNRS, LIS, Marseille, France
Oscar Defrain
  • Aix-Marseille Université, CNRS, LIS, Marseille, France
Frédéric Olive
  • Aix-Marseille Université, CNRS, LIS, Marseille, 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 would like to thank Yasuaki Kobayashi, Kazuhiro Kurita, Kazuhisa Makino and Kunihiro Wasa for preliminary discussions on the topic of this paper.

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