On convex relaxation of graph isomorphism

Significance Graphs are a ubiquitous mathematical abstraction used in numerous problems in science and engineering. Of particular importance is the need to find the best structure-preserving matching of graphs. Since graph matching (GM) is a computationally intractable problem, numerous heuristics exist to approximate its solution. An important class of GM heuristics is relaxation techniques based on replacing the original problem by a continuous convex program. Conditions for applicability or inapplicability of such convex relaxations are poorly understood. In this study, we propose easy to check spectral properties to characterize a wide family of graphs for which we prove equivalence of convex relaxation to the exact GM. We consider the problem of exact and inexact matching of weighted undirected graphs, in which a bijective correspondence is sought to minimize a quadratic weight disagreement. This computationally challenging problem is often relaxed as a convex quadratic program, in which the space of permutations is replaced by the space of doubly stochastic matrices. However, the applicability of such a relaxation is poorly understood. We define a broad class of friendly graphs characterized by an easily verifiable spectral property. We prove that for friendly graphs, the convex relaxation is guaranteed to find the exact isomorphism or certify its inexistence. This result is further extended to approximately isomorphic graphs, for which we develop an explicit bound on the amount of weight disagreement under which the relaxation is guaranteed to find the globally optimal approximate isomorphism. We also show that in many cases, the graph matching problem can be further harmlessly relaxed to a convex quadratic program with only n separable linear equality constraints, which is substantially more efficient than the standard relaxation involving [Formula] equality and [Formula] inequality constraints. Finally, we show that our results are still valid for unfriendly graphs if additional information in the form of seeds or attributes is allowed, with the latter satisfying an easy to verify spectral characteristic.