Limitations of Algebraic Approaches to Graph Isomorphism Testing

TL;DR

This paper examines the limitations of algebraic methods, such as Gr"obner bases and algebraic proof systems, for graph isomorphism testing, establishing lower bounds and comparing with combinatorial algorithms.

Contribution

It provides linear lower bounds on algebraic proof system degrees for graph isomorphism and characterizes the Weisfeiler-Lehman algorithm's power algebraically.

Findings

Linear lower bounds on polynomial calculus degree over all fields of characteristic not 2

Linear lower bounds on Positivstellensatz calculus derivation degree

Characterization of Weisfeiler-Lehman algorithm's power in algebraic terms

Abstract

We investigate the power of graph isomorphism algorithms based on algebraic reasoning techniques like Gr\"obner basis computation. The idea of these algorithms is to encode two graphs into a system of equations that are satisfiable if and only if if the graphs are isomorphic, and then to (try to) decide satisfiability of the system using, for example, the Gr\"obner basis algorithm. In some cases this can be done in polynomial time, in particular, if the equations admit a bounded degree refutation in an algebraic proof systems such as Nullstellensatz or polynomial calculus. We prove linear lower bounds on the polynomial calculus degree over all fields of characteristic different from 2 and also linear lower bounds for the degree of Positivstellensatz calculus derivations. We compare this approach to recently studied linear and semidefinite programming approaches to isomorphism testing, which are known to be related to the combinatorial Weisfeiler-Lehman algorithm. We exactly characterise the power of the Weisfeiler-Lehman algorithm in terms of an algebraic proof system that lies between degree-k Nullstellensatz and degree-k polynomial calculus.