Constructing Hard Examples for Graph Isomorphism

TL;DR

This paper presents a method to generate challenging graph instances for testing graph isomorphism algorithms, based on theoretical constructions from random 3XOR formulas, validated through experiments.

Contribution

It introduces a novel construction technique for hard graph instances using properties of random 3XOR formulas, bridging theory and practical testing.

Findings

Generated graphs are as challenging as the hardest known benchmarks.

The method successfully produces graphs with high Weisfeiler-Leman dimension.

Experimental validation confirms the effectiveness of the approach.

Abstract

We describe a method for generating graphs that provide difficult examples for practical Graph Isomorphism testers. We first give the theoretical construction, showing that we can have a family of graphs without any non-trivial automorphisms which also have high Weisfeiler-Leman dimension. The construction is based on properties of random 3XOR-formulas. We describe how to convert such a formula into a graph which has the desired properties with high probability. We validate the method by an experimental implementation. We construct random formulas and validate them with a SAT solver to filter through suitable ones, and then convert them into graphs. Experimental results demonstrate that the resulting graphs do provide hard examples that match the hardest known benchmarks for graph isomorphism.