On the Largest Common Subgraph Problem

Given two graphs Gi = (Vi, Ei) and G2 = (V2, £2), |Vi| = \V2\ = n, to determine whether they have a size-A; common subgraph is one of the earliest examples of an NP-complete problem (by a trivial reduction from the Maximum Clique problem). We show that this problem for equally sized G\ and G2, i.e. when |23i| = |2?2| = m, remains NP-complete. Moreover, the restriction to the case k = m—tfn, c> 1, is also NP-complete. In this result k and m can hardly be made tighter because the largest Common Subgraph problem for equally sized graphs is reducible to the Graph Isomorphism problem in time 7i°(m-fc). Further, we consider the optimization problem of Computing the maximum common subgraph size. It is only known that this pro blem is not harder than Computing the maximum clique size (V.Kann, STACS'92), and that it is approximable within factor 0( j^tn) (M.Halldörsson, 1994). For some e € (0,1), weprove that the largest common subgraph size is not approximable within addend nc unless NP = P. The techniques used are reductions from the problem of distinguishing between graphs with large and small clique size. •Supported in part by grant No. MGT 000 from the International Science Foundation. Part of this work was done while visiting the Ulm university.