A Separator-based Algorithm for the Graph Edit Distance Problem

2026-08-05

AI Summary

This work addresses the problem of computing the exact graph edit distance (GED) between labeled graphs. It introduces, for the first time, graph separator theory into exact GED computation, specifically targeting cases where one input graph admits a sublinear balanced separator—such as planar graphs or graphs of bounded treewidth. The authors propose an exponential-time algorithm based on separator decomposition and dynamic programming, achieving a running time of O*((4+ε)^n) within polynomial space. This represents a significant improvement over the traditional O*(n!) complexity barrier. Empirical evaluation demonstrates that the method yields exponential speedups over existing worst-case algorithms on real-world graph datasets from benchmarks such as GedLib, particularly for graph classes amenable to efficient separator-based decomposition.

Abstract

The Graph Edit Distance (GED) is a widely used graph similarity measure asking for the minimum cost of a sequence of edits transforming one (labeled) graph into another. The considered edit operations are deletion, insertion, and relabeling of nodes and edges. Special cases include the Graph Isomorphism problem, as well as many other graph problems that ask for the existence or minimum cost of a certain substructure, like the Traveling Salesman or Maximum Clique problem. We present a novel exponential time algorithm to compute the exact GED and a corresponding edit sequence in $O^*(4 + \varepsilon)^n$ time and polynomial space, provided one of the two graphs admits strictly sublinear balanced separators. In particular, the claimed runtime holds if one of the graphs is $K_h$-minor free (e.g., planar), or has bounded treewidth, which is the case for many real-world applications (e.g., all instances in GEDLIB). This substantially improves the best known worst-case running time bounds of $O^*(n!)$ for these graph classes.

Problem

• Graph Edit Distance
• Graph Similarity
• Graph Isomorphism
• Combinatorial Optimization
• Exact Algorithm

Innovation

• Graph Edit Distance
• balanced separators
• exponential-time algorithm
• bounded treewidth
• K_h-minor free