Graph Homomorphism Revisited for Graph Matching

In a variety of emerging applications one needs to decide whethera graph G matches another Gp, i.e., whether G has a topologicalstructure similar to that of Gp. The traditional notions of graphhomomorphism and isomorphism often fall short of capturing thestructural similarity in these applications. This paper studies revisions of these notions, providing a full treatment from complexityto algorithms. (1) We propose p-homomorphism (p-hom) and 1-1p-hom, which extend graph homomorphism and subgraph isomor-phism, respectively, by mapping edges from one graph to paths inanother, and by measuring the similarity of nodes. (2) We intro-duce metrics to measure graph similarity, and several optimizationproblems for p-hom and 1-1 p-hom. (3) We show that the deci-sion problems for p-hom and 1-1 p-hom are NP-complete even forDAGs, and that the optimization problems are approximation-hard.(4) Nevertheless, we provide approximation algorithms with prov-able guarantees on match quality. We experimentally verify theeffectiveness of the revised notions and the efficiency of our algo-rithms in Web site matching, using real-life and synthetic data.