Robust Graph Isomorphism, Quadratic Assignment and VC Dimension
For researchers in graph algorithms and machine learning, this work provides efficient approximation algorithms for fundamental problems on graphs with bounded VC dimension, improving over previous results.
The paper presents an additive εn²-approximation algorithm for Graph Edit Distance on graphs with bounded VC dimension, running in time n^{O(d/ε²)}, and extends this to Quadratic Assignment Problems. It also shows that Weisfeiler-Leman with dimension O(ε⁻¹d log(ε⁻¹)) solves ε-GI on VC dimension d graphs.
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 $Ω(n^{2}/k)$.