Lov\'asz Meets Weisfeiler and Leman

TL;DR

This paper connects Lovász's graph theory with the Weisfeiler-Leman algorithm, characterizing graph indistinguishability through homomorphism counts, linear systems, and relaxations, leading to efficient isomorphism testing methods.

Contribution

It establishes a deep theoretical link between graph homomorphisms, color refinement, and linear programming relaxations, extending understanding of graph isomorphism algorithms.

Findings

Graphs indistinguishable by color refinement have identical homomorphism counts from all trees.

Nonnegative real solutions to the linear system indicate fractional isomorphisms.

The results lead to a quasi-linear time algorithm for certain graph isomorphism problems.

Abstract

In this paper, we relate a beautiful theory by Lov\'asz with a popular heuristic algorithm for the graph isomorphism problem, namely the color refinement algorithm and its k-dimensional generalization known as the Weisfeiler-Leman algorithm. We prove that two graphs G and H are indistinguishable by the color refinement algorithm if and only if, for all trees T, the number Hom(T,G) of homomorphisms from T to G equals the corresponding number Hom(T,H) for H. There is a natural system of linear equations whose nonnegative integer solutions correspond to the isomorphisms between two graphs. The nonnegative real solutions to this system are called fractional isomorphisms, and two graphs are fractionally isomorphic if and only if the color refinement algorithm cannot distinguish them (Tinhofer 1986, 1991). We show that, if we drop the nonnegativity constraints, that is, if we look for arbitrary real solutions, then a solution to the linear system exists if and only if, for all t, the two graphs have the same number of length-t walks. We lift the results for trees to an equivalence between numbers of homomorphisms from graphs of tree width k, the k-dimensional Weisfeiler-Leman algorithm, and the level-k Sherali-Adams relaxation of our linear program. We also obtain a partial result for graphs of bounded path width and solutions to our system where we drop the nonnegativity constraints. A consequence of our results is a quasi-linear time algorithm to decide whether, for two given graphs G and H, there is a tree T with Hom(T,G) = Hom(T,H).