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On the Relative Power of Linear Algebraic Approximations of Graph Isomorphism

Authors Anuj Dawar , Danny Vagnozzi

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Author Details

Anuj Dawar
  • Department of Computer Science and Technology, University of Cambridge, UK
Danny Vagnozzi
  • Department of Computer Science and Technology, University of Cambridge, UK


We want to thank Martin Grohe, Benedikt Pago and Gregory Wilsenach for useful discussions.

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Anuj Dawar and Danny Vagnozzi. On the Relative Power of Linear Algebraic Approximations of Graph Isomorphism. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 37:1-37:16, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)


We compare the capabilities of two approaches to approximating graph isomorphism using linear algebraic methods: the invertible map tests (introduced by Dawar and Holm) and proof systems with algebraic rules, namely polynomial calculus, monomial calculus and Nullstellensatz calculus. In the case of fields of characteristic zero, these variants are all essentially equivalent to the Weisfeiler-Leman algorithms. In positive characteristic we show that the distinguishing power of the monomial calculus is no greater than the invertible map method by simulating the former in a fixed-point logic with solvability operators. In turn, we show that the distinctions made by this logic can be implemented in the Nullstellensatz calculus.

Subject Classification

ACM Subject Classification
  • Theory of computation → Finite Model Theory
  • Theory of computation → Proof complexity
  • Theory of computation → Complexity theory and logic
  • Graph isomorphism
  • proof complexity
  • invertible map tests


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