# Convex Graph Invariant Relaxations For Graph Edit Distance

## TL;DR

This paper introduces a new family of convex relaxations that provide lower bounds for graph edit distance, leveraging graph invariants to improve computational efficiency and accuracy.

## Contribution

It proposes a novel framework of convex relaxations using graph invariants to estimate graph edit distance more effectively.

## Key findings

- Relaxations can be tight when one graph has few eigenvalues.
- Framework is validated on synthetic and molecular data.
- Provides lower bounds, complementing existing upper-bound heuristics.

## Abstract

The edit distance between two graphs is a widely used measure of similarity that evaluates the smallest number of vertex and edge deletions/insertions required to transform one graph to another. It is NP-hard to compute in general, and a large number of heuristics have been proposed for approximating this quantity. With few exceptions, these methods generally provide upper bounds on the edit distance between two graphs. In this paper, we propose a new family of computationally tractable convex relaxations for obtaining lower bounds on graph edit distance. These relaxations can be tailored to the structural properties of the particular graphs via convex graph invariants. Specific examples that we highlight in this paper include constraints on the graph spectrum as well as (tractable approximations of) the stability number and the maximum-cut values of graphs. We prove under suitable conditions that our relaxations are tight (i.e., exactly compute the graph edit distance) when one of the graphs consists of few eigenvalues. We also validate the utility of our framework on synthetic problems as well as real applications involving molecular structure comparison problems in chemistry.