On the unification of the graph edit distance and graph matching problems

Abstract

Error-tolerant graph matching gathers an important family of problems. These problems aim at finding correspondences between two graphs while integrating an error model. In the Graph Edit Distance (GED) problem, the insertion/deletion of edges/nodes from one graph to another is explicitly expressed by the error model. At the opposite, the problem commonly referred to as "graph matching" does not explicitly express such operations. For decades, these two problems have split the research community in two separated parts. It resulted in the design of different solvers for the two problems. In this paper, we propose a unification of both problems thanks to a single model. We give the proof that the two problems are equivalent under a reformulation of the error models. This unification makes possible the use on both problems of existing solving methods from the two communities.

Graphs in pattern recognition: successes, shortcomings, and perspectives

“…The latter can be stated as follows: given two graphs and some edit operations with an edit cost, GED is the problem of finding the minimum sum of edit cost to transform a graph into the other; GED also provides a node-to-node correspondence between the nodes of the two graphs. Actually in 2021, Raveaux 16 shows that the GED problem can be equivalent to the GM problem under certain permissive conditions. Nevertheless, as GED has been widely addressed in the literature, we propose here some of the key papers on this specific problem.…”

SIGMA: A Structural Inconsistency Reducing Graph Matching Algorithm

“…Spectral distances usually do not take into account all the structural information, focusing only on the Laplacian matrix eigenvectors and ignoring a large portion of the structure encoded in eigenvectors [Jovanović andStanić, 2012, Gera et al, 2018]. The cut distance [Lovász, 2012] and the graph edit distance [Bougleux et al, 2017, Raveaux, 2021 require solving difficult discrete optimization problems. Another family of graph distances that is closely related to this paper is the graph diffusion distance [Hammond et al, 2013, Tsitsulin et al, 2018, Scott and Mjolsness, 2021 8) also defines a distance between two graphs.…”

Deep graph matching meets mixed-integer linear programming: Relax at your own risk ?

“…In [14], a MILP is proposed to model the graph edit distance problem that is another graph matching problem. However, graph matching and graph edit distance problems have been unified in [33]. The graph edit distance problem is a cost minimization problem while the graph matching problem (Problem 1) is similarity maximization problem.…”

“…The graph edit distance problem is a cost minimization problem while the graph matching problem (Problem 1) is similarity maximization problem. Consequently as proven in [33], the MILP proposed in [14] can model the graph matching problem by changing the MILP from a minimization to a maximization problem. The MILP has been showed to be efficient because it is compact in terms of variables and constraints.…”