Robust Graph Isomorphism, Quadratic Assignment and VC Dimension

2026-04-14

AI Summary

This work investigates efficient approximation algorithms for graph edit distance (GED), the quadratic assignment problem (QAP), and ε-approximate graph isomorphism (ε-GI) on graphs of bounded VC dimension. For graphs with VC dimension d, it presents the first additive εn²-approximation algorithm grounded in VC dimension theory, improving the running times for GED and weighted QAP to n^{O(d/ε²)} and n^{O(ε⁻²(d + log ε⁻¹))}, respectively. Moreover, it establishes the first theoretical connection between the Weisfeiler–Leman (WL) dimension and both the VC dimension and the tolerance parameter ε, proving that O(ε⁻¹d log ε⁻¹)-dimensional WL can decide ε-GI. These results substantially advance the state of the art in approximation complexity for these fundamental problems and extend the applicability of VC dimension theory to combinatorial optimization.

Technology Category

• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Machine Learning: Graph-based Machine Learning
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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 $Ω(n^{2}/k)$.

Problem

• Graph Edit Distance
• Graph Isomorphism
• Quadratic Assignment Problem
• VC Dimension
• Weisfeiler-Leman Algorithm

Innovation

• Graph Edit Distance
• VC Dimension
• Quadratic Assignment Problem
• Weisfeiler–Leman Algorithm
• Robust Graph Isomorphism