The approximate graph matching problem

Labeled graphs are graphs in which each node and edge has a label. The distance between two labeled graphs is considered to be the weighted sum of the costs of edit operations (insert, delete and relabel the nodes and edges) to transform one graph to the other. The paper considers two variants of the approximate graph matching (AGM) problem: given a pattern graph P and a data graph D, what is the distance between P and D? and what is the minimum distance between P and D when subgraphs can be freely removed from D? We show that no efficient algorithm can solve either variant of the AGM, unless P=NP. We then give a polynomial-time approximation algorithm to solve this problem.