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 $Ω(n^{2}/k)$.

翻译：我们针对VC维为$d$的图，提出一种加性$\varepsilon n^{2}$近似的图编辑距离问题（GED）算法，运行时间为$n^{O(d/\varepsilon^{2})}$。这一结果推广了Arora、Frieze和Kaplan先前的工作[Math. Program. 2002]，他们给出的$\varepsilon n^{2}$近似算法运行时间为$n^{O(\log n/\varepsilon^{2})}$。与Arora等人的工作类似，我们通过引入QAP实例的VC维概念，将结果拓展至任意二次分配问题（QAP），并给出有界权重QAP的$\varepsilon n^{2}$近似算法，运行时间为$n^{O(\varepsilon^{-2}(d + \log\varepsilon^{-1}))}$。作为一个特别有意义的特例，我们进一步研究$\varepsilon$-$\mathsf{GI}$问题：给定两个$n$顶点图$G,H$，需判定它们是否同构，并承诺若不同构，则其图编辑距离至少为$\varepsilon n^{2}$。我们证明，维数为$O(\varepsilon^{-1}d\log(\varepsilon^{-1}))$的标准Weisfeiler-Leman算法可解决VC维为$d$的图上的该问题。同时，对于任意$n$顶点图，维数$O(\varepsilon^{-1}\log n)$即足够，而$k$-WL算法在距离为$Ω(n^{2}/k)$的实例上则会失效。