Resolution with Symmetry Rule Applied to Linear Equations

Schweitzer, Pascal; Seebach, Constantin

doi:10.4230/lipics.stacs.2021.58

Cited by 3 publications

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“…An alternative set of affine inequalities over the same set of variables is sometimes more convenient and has been used before for encoding the isomorphism principle in other proof systems like Resolution [51,48,52] or Polynomial Calculus [6]. Instead of the inequalities for the matrices, for every two pairs of vertices v, v ′ ∈ V G and w, w ′ ∈ V H such that (v, v ′ ) is an edge in G and (w, w ′ ) is not an edge in H (or the other way around) we include an inequality indicating that v is not mapped to w or v ′ is not mapped to w ′ .…”

Section: Two Sets Of Affine Inequalities For Graph Isomorphismmentioning

confidence: 99%

Cutting Planes Width and the Complexity of Graph Isomorphism Refutations

Torán,

Wörz

2024

ACM Trans. Comput. Logic

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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 used method for proving size lower bounds, and it is known that for the mentioned proof 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 2009 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 \(\mathsf{WL}(G,H)\) of the graphs and the width of a CP refutation for the corresponding isomorphism formula Iso \((G,H)\) . In particular, we show that if \(\mathsf{WL}(G,H)\leq k\) , then there is a CP refutation of Iso \((G,H)\) with width \(k\) , and if \(\mathsf{WL}(G,H){\gt}\,k\) , 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-length CP refutations from our width bound for isomorphism formulas for graphs with constant Weisfeiler–Leman dimension. Furthermore, we notice that a length lower bound for refuting graph isomorphism formulas in the subsystem of tree-like Cutting Planes with polynomially bounded coefficients follows from known results.

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Section: Two Sets Of Affine Inequalities For Graph Isomorphismmentioning

confidence: 99%

Cutting Planes Width and the Complexity of Graph Isomorphism Refutations

Torán,

Wörz

2024

ACM Trans. Comput. Logic

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Number of Variables for Graph Differentiation and the Resolution of Graph Isomorphism Formulas

Torán

Wörz

2023

ACM Trans. Comput. Logic

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We show that the number of variables and the quantifier depth needed to distinguish a pair of graphs by first-order logic sentences exactly match the complexity measures of clause width and depth needed to refute the corresponding graph isomorphism formula in propositional narrow resolution. Using this connection, we obtain upper and lower bounds for refuting graph isomorphism formulas in (normal) resolution. In particular, we show that if k is the minimum number of variables needed to distinguish two graphs with n vertices each, then there is an n O( k ) resolution refutation size upper bound for the corresponding isomorphism formula, as well as lower bounds of 2 k − 1 and k for the tree-like resolution size and resolution clause space for this formula. We also show a (normal) resolution size lower bound of exp ( Ω ( k 2 / n )) for the case of colored graphs with constant color class sizes. Applying these results, we prove the first exponential lower bound for graph isomorphism formulas in the proof system SRC-1, a system that extends resolution with a global symmetry rule, thereby answering an open question posed by Schweitzer and Seebach.

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Number of Variables for Graph Differentiation and the Resolution of GI Formulas

Torán,

Wörz

2022

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Resolution with Symmetry Rule Applied to Linear Equations

Cited by 3 publications

References 0 publications

Cutting Planes Width and the Complexity of Graph Isomorphism Refutations

Cutting Planes Width and the Complexity of Graph Isomorphism Refutations

Number of Variables for Graph Differentiation and the Resolution of Graph Isomorphism Formulas

Number of Variables for Graph Differentiation and the Resolution of GI Formulas

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