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Cutting Planes Width and the Complexity of Graph Isomorphism Refutations

Authors Jacobo Torán , Florian Wörz

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LIPIcs.SAT.2023.26.pdf
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Jacobo Torán
  • Institut für Theoretische Informatik, Universität Ulm, Germany
Florian Wörz
  • Institut für Theoretische Informatik, Universität Ulm, Germany

Funding

Supported by the Deutsche Forschungsgemeinschaft (DFG) under project number 430150230, "Complexity measures for solving propositional formulas".

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Jacobo Torán and Florian Wörz. Cutting Planes Width and the Complexity of Graph Isomorphism Refutations. In 26th International Conference on Theory and Applications of Satisfiability Testing (SAT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 271, pp. 26:1-26:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023) https://doi.org/10.4230/LIPIcs.SAT.2023.26

Abstract

The width complexity measure plays a central role in Resolution and other propositional proof systems like Polynomial Calculus (under the name of degree). The study of width lower bounds is the most extended method for proving size lower bounds, and it is known that for these systems, proofs with small width also imply the existence of proofs with small size. Not much has been studied, however, about the width parameter in the Cutting Planes (CP) proof system, a measure that was introduced by Dantchev and Martin in 2011 under the name of CP cutwidth.
In this paper, we study the width complexity of CP refutations of graph isomorphism formulas. For a pair of non-isomorphic graphs G and H, we show a direct connection between the Weisfeiler-Leman differentiation number WL(G, H) of the graphs and the width of a CP refutation for the corresponding isomorphism formula Iso(G, H). In particular, we show that if WL(G, H) ≤ k, then there is a CP refutation of Iso(G, H) with width k, and if WL(G, H) > k, then there are no CP refutations of Iso(G, H) with width k-2. Similar results are known for other proof systems, like Resolution, Sherali-Adams, or Polynomial Calculus. We also obtain polynomial-size CP refutations from our width bound for isomorphism formulas for graphs with constant WL-dimension.

Subject Classification

ACM Subject Classification
  • Theory of computation → Proof complexity
  • Theory of computation → Complexity theory and logic
  • Mathematics of computing → Graph theory
Keywords
  • Cutting Planes
  • Proof Complexity
  • Linear Programming
  • Combinatorial Optimization
  • Weisfeiler-Leman Algorithm
  • Graph Isomorphism

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