A Path Following Algorithm for the Graph Matching Problem

TLDR

The concave relaxation shares the same global minimum as the original graph matching problem, yet finding that minimum remains a hard combinatorial challenge. We propose a convex‑concave programming approach to solve the labeled weighted graph matching problem. We reformulate the weighted graph matching as a least‑square problem over permutation matrices, relax it to convex and concave quadratic programs on doubly stochastic matrices, follow a solution path from the convex to the concave formulation, and evaluate the resulting algorithm against leading methods on simulated graphs, QAPLib, retina vessel images, and handwritten Chinese characters. The method incorporates graph label similarities into the optimization and achieves results competitive with the state of the art on all tested datasets.

Abstract

We propose a convex-concave programming approach for the labeled weighted graph matching problem. The convex-concave programming formulation is obtained by rewriting the weighted graph matching problem as a least-square problem on the set of permutation matrices and relaxing it to two different optimization problems: a quadratic convex and a quadratic concave optimization problem on the set of doubly stochastic matrices. The concave relaxation has the same global minimum as the initial graph matching problem, but the search for its global minimum is also a hard combinatorial problem. We, therefore, construct an approximation of the concave problem solution by following a solution path of a convex-concave problem obtained by linear interpolation of the convex and concave formulations, starting from the convex relaxation. This method allows to easily integrate the information on graph label similarities into the optimization problem, and therefore, perform labeled weighted graph matching. The algorithm is compared with some of the best performing graph matching methods on four data sets: simulated graphs, QAPLib, retina vessel images, and handwritten Chinese characters. In all cases, the results are competitive with the state of the art.