Spectral Clustering for Divide-and-Conquer Graph Matching

Graph matching is an increasingly important problem in inferential graph statistics, with applications across a broad spectrum of fields including computer vision ([38], [10]), shape matching and object recognition ([4], [7]), and biology and neuroscience ([22], [34], [37]), to name a few. The graph matching problem (GMP) seeks to find an alignment between the vertex sets of two graphs that best preserves common structure across graphs. Unfortunately, the GMP is inherently combinatorial, and no efficient exact graph matching algorithms are known. Indeed, even the simpler problem of determining if two graphs are isomorphic is famously of unknown complexity ([19], 1 [30]), and if the graphs are allowed to be loopy, weighted and directed, then the simplest version of GMP is equivalent to the NPhard quadratic assignment problem. Due to its wide applicability, there exist a vast number of approximating algorithms for GMP; see the paper "30 Years of Graph Matching in Pattern Recognition" ([11]) for an excellent survey of the existing literature.