Optimal graph joining with applications to isomorphism detection and identification

TL;DR

This paper presents an optimal transport-based method for comparing undirected graphs that leverages linear programming to address graph isomorphism detection, establishing theoretical links and conditions for accurate identification.

Contribution

It introduces the optimal graph joining (OGJ) framework, connecting optimal transport with graph isomorphism, and provides theoretical conditions for isomorphism detection using linear programming.

Findings

OGJ problem can be formulated as a linear program with convex solutions.

Conditions are identified under which the optimal joining cost indicates graph isomorphism.

The solution set's extreme points correspond to isomorphisms when certain conditions are met.

Abstract

We introduce an optimal transport based approach for comparing undirected graphs with non-negative edge weights and general vertex labels, and we study connections between the resulting linear program and the graph isomorphism problem. Our approach is based on the notion of a joining of two graphs $G$ and $H$, which is a product graph that preserves their marginal structure. Given $G$ and $H$ and a vertex-based cost function $c$, the optimal graph joining (OGJ) problem finds a joining of $G$ and $H$ minimizing degree weighted cost. The OGJ problem can be written as a linear program with a convex polyhedral solution set. We establish several basic properties of the OGJ problem, and present theoretical results connecting the OGJ problem to the graph isomorphism problem. In particular, we examine a variety of conditions on graph families that are sufficient to ensure that for every pair of graphs $G$ and $H$ in the family (i) $G$ and $H$ are isomorphic if and only if their optimal joining cost is zero, and (ii) if $G$ and $H$ are isomorphic, the the extreme points of the solution set of the OGJ problem are deterministic joinings corresponding to the isomorphisms from $G$ to $H$.