Robust Graph Isomorphism, Quadratic Assignment and VC Dimension

TL;DR

This paper develops approximation algorithms for graph edit distance and quadratic assignment problems on graphs with bounded VC dimension, and analyzes the Weisfeiler--Leman algorithm's effectiveness in graph isomorphism testing.

Contribution

It introduces VC dimension-based approximation algorithms for GED and QAP, and characterizes the Weisfeiler--Leman algorithm's capabilities for graph isomorphism under these conditions.

Findings

Provides an additive $ ext{ extdollar} ext{ extbackslash}varepsilon n^{2}$-approximation for GED on VC dimension $d$ graphs.

Extends approximation results to quadratic assignment problems with bounded weights.

Shows Weisfeiler--Leman algorithm solves $ ext{ extdollar} ext{ extbackslash}varepsilon$-GI on graphs of VC dimension $d$.

Abstract

We present an additive $\varepsilon n^{2}$-approximation algorithm for the Graph Edit Distance problem (GED) on graphs of VC dimension $d$ running in time $n^{O(d/\varepsilon^{2})}$. In particular, this recovers a previous result by Arora, Frieze, and Kaplan [Math. Program. 2002] who gave an $\varepsilon n^{2}$-approximation running in time $n^{O(\log n/\varepsilon^{2})}$. Similar to the work of Arora et al., we extend our results to arbitrary Quadratic Assignment problems (QAPs) by introducing a notion of VC dimension for QAP instances, and giving an $\varepsilon n^{2}$-approximation for QAPs with bounded weights running in time $n^{O(\varepsilon^{-2}(d + \log\varepsilon^{-1}))}$. As a particularly interesting special case, we further study the problem $\varepsilon$-$\mathsf{GI}$, which entails determining if two graphs $G,H$ over $n$ vertices are isomorphic, when promised that if they are not, their graph edit distance is at least $\varepsilon n^{2}$. We show that the standard Weisfeiler--Leman algorithm of dimension $O(\varepsilon^{-1}d\log(\varepsilon^{-1}))$ solves this problem on graphs of VC dimension $d$. We also show that dimension $O(\varepsilon^{-1}\log n)$ suffices on arbitrary $n$-vertex graphs, while $k$-WL fails on instances at distance $\Omega(n^{2}/k)$.