Graph homomorphism revisited for graph matching

In a variety of emerging applications one needs to decide whether a graph G matches another G_p, i.e., whether G has a topological structure similar to that of G_p. The traditional notions of graph homomorphism and isomorphism often fall short of capturing the structural similarity in these applications. This paper studies revisions of these notions, providing a full treatment from complexity to algorithms. (1) We propose p-homomorphism (p-hom) and 1-1 p-hom, which extend graph homomorphism and subgraph isomorphism, respectively, by mapping edges from one graph to paths in another, and by measuring the similarity of nodes. (2) We introduce metrics to measure graph similarity, and several optimization problems for p-hom and 1-1 p-hom. (3) We show that the decision problems for p-hom and 1-1 p-hom are NP-complete even for DAGs, and that the optimization problems are approximation-hard. (4) Nevertheless, we provide approximation algorithms with provable guarantees on match quality. We experimentally verify the effectiveness of the revised notions and the efficiency of our algorithms in Web site matching, using real-life and synthetic data.