On the relative power of algebraic approximations of graph isomorphism

TL;DR

This paper compares algebraic methods for graph isomorphism approximation, analyzing their relative power and potential for extension, especially in different field characteristics, to better understand their capabilities and limitations.

Contribution

It establishes the equivalence of certain algebraic approaches to Weisfeiler-Leman algorithms in characteristic zero and explores their relationships in positive characteristic.

Findings

Invertible map tests are equivalent to Weisfeiler-Leman in characteristic zero.

Invertible map method can simulate monomial calculus in positive characteristic.

Potential extensions of invertible map tests to monomial calculus are identified.

Abstract

We compare the capabilities of two approaches to approximating graph isomorphism using linear algebraic methods: the \emph{invertible map tests} (introduced by Dawar and Holm) and proof systems with algebraic rules, namely \emph{polynomial calculus}, \emph{monomial calculus} and \emph{Nullstellensatz calculus}. In the case of fields of characteristic zero, these variants are all essentially equivalent to the the Weisfeiler-Leman algorithms. In positive characteristic we show that the invertible map method can simulate the monomial calculus and identify a potential way to extend this to the monomial calculus.