Graph homomorphism revisited for graph matching

Abstract

In a variety of emerging applications one needs to decide whether a graph G matches another Gp, i.e., whether G has a topological structure similar to that of Gp. 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.

Foundations of Modern Query Languages for Graph Databases

“…The evaluation of Q against G then consists of all possible homomorphisms from Q to G. Since homomorphism-based approach corresponds to the familiar semantics of select-from-where queries in relational databases, and since it forms the basis for the other, more restrictive semantics of bgps, it is often studied in the theoretical community (see, e.g., [Calvanese et al 2000;Wood 2012;). There are also several papers that study implementation issues related to this semantics (see, e.g., [Cheng et al 2008;Zou et al 2009;Fan et al 2010b]) and it is currently used, for example, by the SPARQL query language [Harris and Seaborne 2013]. (2) Isomorphism-based semantics: Under this type of semantics, the structure of the query (in some potentially application-dependent sense) should be preserved under the image of the permitted mappings; in more practical terms, certain types of variables are restricted to match distinct constants in the database.…”

“…Then an answer to Q over G under the simulation-based semantics is any simulation S between Q and G. As shown in the literature, simulation-based semantics is computationally lighter for certain problems [Henzinger et al 1995;Fan et al 2011] and is more versatile when handling large graphs that might contain incomplete information [Fan et al 2010a;Fan 2012;Fan et al 2010b]. …”

Virtual Network Mapping: A Graph Pattern Matching Approach

“…However, those results are for edge-to-edge mappings, whereas VNM typically needs to map virtual links to physical paths. There have been recent extensions to support edge-to-path mappings for graph pattern matching [16,17], with several intractability and approximation bounds established there. Those differ from this work in that either no constraints on links are considered [17], or graph simulation is adopted [16], which does not work for VNM.…”

“…There have been recent extensions to support edge-to-path mappings for graph pattern matching [16,17], with several intractability and approximation bounds established there. Those differ from this work in that either no constraints on links are considered [17], or graph simulation is adopted [16], which does not work for VNM. The complexity and approximation bounds developed in this work are among the first results that have been developed for VNM in cloud computing.…”

LaSaS: an Aggregated Search based Graph Matching Approach

“…Another variant of inexact graph matching is the graph homomorphism [13]. The idea is to find mappings to the query such that node labels difference falls under a threshold whereas the query edges are mapped to paths of a given length.…”