Robust Graph Isomorphism, Quadratic Assignment and VC Dimension

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 $Ω(n^{2}/k)$.

• The paper establishes a rigorous connection between VC dimension and tractable approximations for GED, QAP, and robust graph isomorphism.
• It employs constructive ε-net techniques to achieve additive O(ε n²)-approximations with subexponential dependence on the VC dimension.
• The study refines Weisfeiler-Leman bounds, demonstrating optimal WL dimension thresholds for distinguishing graph isomorphism in dense and bounded VC regimes.

Robust Graph Isomorphism, Quadratic Assignment, and VC Dimension

Introduction and Problem Setting

This paper establishes a rigorous connection between the VC dimension of input structures and the algorithmic tractability of core combinatorial problems, notably the Graph Edit Distance (GED), the Quadratic Assignment Problem (QAP), and robust versions of the Graph Isomorphism (GI) problem. The results yield additive $O(\varepsilon n^2)$-approximation algorithms for GED and QAP on instances of bounded VC dimension with subexponential dependence on the combinatorial complexity parameter $d$ (the VC dimension). The authors further analyze robust GI under large edit distance promises, showing optimal Weisfeiler-Leman (WL) dimension bounds for distinguishing non-isomorphic graphs in dense and bounded VC dimension regimes.

The approach generalizes and sharpens classical analyses for dense graph problems, leveraging VC dimension as a unifying parameter that controls the sample complexity of $\varepsilon$-nets and impacts the efficiency of combinatorial and algebraic graph algorithms.

Additive Approximation for Graph Edit Distance via VC Dimension

Given two $n$-vertex graphs $G$ and $H$, the GED is the minimum number of edge modifications needed to transform $G$ into an isomorphic copy of $H$. The problem is NP-hard (already for trees) and admits no PTAS unless $\mathbf{P} = \mathbf{NP}$. Prior work by Arora, Frieze, and Kaplan gave an additive $O(n^2)$-approximation in time $d$0 for dense graphs. This paper's main technical advance is demonstrating that if the input graphs have neighbourhood set systems of VC dimension at most $d$1, the same additive approximation can be attained in time $d$2.

The algorithmic core constructs explicit $d$3-nets in the space of neighbourhoods, whose size and sampling complexity are $d$4. This yields polynomial-sized nets for constant $d$5, making the existential arguments constructive. The reduction to VC dimension further extends to edge-weighted graphs using threshold set systems, with an analogous approximation guarantee.

Quadratic Assignment Problem: VC Generalization and Algorithmic Implications

The QAP encompasses a broad family of combinatorial optimization formulations, where the cost function can be viewed as a sum over all pairs under a bipartite assignment, parameterized by a coefficient tensor $d$6. The authors introduce a notion of VC dimension adapted to QAP by considering the induced weighted (or thresholded) set systems on $d$7. For bounded VC dimension $d$8, coefficients bounded by $d$9, and a fixed error $\varepsilon$0, they provide an $\varepsilon$1 additive approximation in time $\varepsilon$2.

The reduction from GED to QAP preserves the relevant VC dimension parameter up to a constant factor. In the special case where the coefficients take few values, the dependence on the value set cardinality substitutes for the logarithmic term.

For higher-degree assignment problems (degree $\varepsilon$3), the analysis lifts by induction, maintaining the VC-based sampling guarantees.

Robust Graph Isomorphism and Weisfeiler-Leman Algorithm

The robust GI problem, denoted $\varepsilon$4-GI, asks to distinguish isomorphic pairs from pairs with edit distance at least $\varepsilon$5. The authors demonstrate that the dimensionality of the Weisfeiler-Leman (WL) refinement required to distinguish such pairs collapses from the classical $\varepsilon$6 bound (Cai-Fürer-Immerman) to $\varepsilon$7 for graphs of VC dimension $\varepsilon$8, and to $\varepsilon$9 for arbitrary $n$0-vertex graphs. For colored graphs with maximum color class size $n$1, the required dimension is $n$2.

The analysis is based on the construction of $n$3-nets for the set system of mixed neighborhoods and shows that refraining from shattering large classes suffices for the combinatorial refinement to succeed. This matches the performance of the combinatorially simpler color refinement approach. Notably, the upper bounds are tight: for every $n$4, there exist hard pairs (via blowups of Cai-Fürer-Immerman or O'Donnell-Wright-Wu-Zhou constructions) with quadratic edit distance yet requiring $n$5-WL for distinction, with $n$6.

Theoretical and Practical Ramifications

These results position VC dimension as a master parameter controlling the complexity of edit distance and assignment problems in both unweighted and weighted settings. The findings subsume and generalize previous bounds in dense regimes, showing tractable approximation and isomorphism testing on broad families of graphs, including those with bounded degree, treewidth, clique-width, twin-width, or merge-width.

The WL analysis highlights new algorithmic thresholds for GI under robust promises. Although robust GI remains open for polynomial time for all fixed $n$7, the established WL bounds delineate the landscape sharply and suggest that further progress will require new techniques beyond combinatorial partitioning.

On the QAP side, the structural results suggest that for practical instances arising in combinatorial optimization and computer vision—where the underlying neighborhood set system is often low-complexity—effective additive approximation is algorithmically feasible.

Conclusion

The paper delivers a comprehensive and technically sophisticated treatment of graph edit distance, quadratic assignment, and robust isomorphism in relation to the VC dimension of the instance space. Additive approximation algorithms with explicit complexity bounds are presented for both GED and QAP, with generalizations to higher-order problems. For robust graph isomorphism, the WL dimension is shown to be optimal and polynomial for bounded VC graphs, contrasting sharply with the classical exponential lower bounds for the general GI problem.

Future research directions include closing the gap for robust GI in polynomial time for all $n$8 and extending the VC dimension framework to broader classes of optimization and parameterized problems in graph theory and combinatorics. The techniques developed underscore the utility of VC-theoretic tools in algorithmic graph theory and suggest promising avenues for the design and analysis of efficient algorithms under structural graph constraints.

Reference: "Robust Graph Isomorphism, Quadratic Assignment and VC Dimension" (2604.12584)