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A SAT Solver’s Opinion on the Erdős-Faber-Lovász Conjecture

Authors Markus Kirchweger , Tomáš Peitl, Stefan Szeider

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Author Details

Markus Kirchweger
  • Algorithms and Complexity Group, TU Wien, Austria
Tomáš Peitl
  • Algorithms and Complexity Group, TU Wien, Austria
Stefan Szeider
  • Algorithms and Complexity Group, TU Wien, Austria

Funding

This work was supported by the Austrian Science Fund (FWF, project P32441), and the Vienna Science and Technology Fund (WWTF, project ICT19-065).

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Markus Kirchweger, Tomáš Peitl, and Stefan Szeider. A SAT Solver’s Opinion on the Erdős-Faber-Lovász Conjecture. In 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 271, pp. 13:1-13:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SAT.2023.13

Abstract

In 1972, Paul Erdős, Vance Faber, and Lászlo Lovász asked whether every linear hypergraph with n vertices can be edge-colored with n colors, a statement that has come to be known as the EFL conjecture. Erdős himself considered the conjecture as one of his three favorite open problems, and offered increasing money prizes for its solution on several occasions. A proof of the conjecture was recently announced, for all but a finite number of hypergraphs. In this paper we look at some of the cases not covered by this proof. We use SAT solvers, and in particular the SAT Modulo Symmetries (SMS) framework, to generate non-colorable linear hypergraphs with a fixed number of vertices and hyperedges modulo isomorphisms. Since hypergraph colorability is NP-hard, we cannot directly express in a propositional formula that we want only non-colorable hypergraphs. Instead, we use one SAT (SMS) solver to generate candidate hypergraphs modulo isomorphisms, and another to reject them by finding a coloring. Each successive candidate is required to defeat all previous colorings, whereby we avoid having to generate and test all linear hypergraphs. Computational methods have previously been used to verify the EFL conjecture for small hypergraphs. We verify and extend these results to larger values and discuss challenges and directions. Ours is the first computational approach to the EFL conjecture that allows producing independently verifiable, DRAT proofs.

Subject Classification

ACM Subject Classification
  • Theory of computation → Automated reasoning
  • Mathematics of computing → Hypergraphs
  • Mathematics of computing → Graph coloring
  • Mathematics of computing → Graph enumeration
  • Mathematics of computing → Matchings and factors
Keywords
  • hypergraphs
  • graph coloring
  • SAT modulo symmetries

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