The Maximum Common Subgraph Problem: Faster Solutions via Vertex Cover

In the maximum common subgraph (MCS) problem, we are given a pair of graphs and asked to find the largest induced subgraph common to them both. With its plethora of applications, MCS is a familiar and challenging problem. Many algorithms exist that can deliver optimal MCS solutions, but whose asymptotic worst-case run times fail to do better than mere brute-force, which is exponential in the order of the smaller graph. In this paper, we present a faster solution to MCS. We transform an essential part of the search process into the task of enumerating maximal independent sets in only a part of only one of the input graphs. This is made possible by exploiting an efficient decomposition of a graph into a minimum vertex cover and the maximum independent set in its complement. The result is an algorithm whose run time is bounded by a function exponential in the order of the smaller cover rather than in the order of the smaller graph.