Maximum common edge subgraph

Given two graphs $$G$$ and $$G'$$, the maximum common edge subgraph problem is the problem of finding a graph $$H$$ with as many edges as possible which is isomorphic to both a subgraph of $$G$$ and a subgraph of $$G'$$.

The maximum common edge subgraph problem on general graphs is NP-complete as it is a generalization of subgraph isomorphism: a graph $$H$$ is isomorphic to a subgraph of another graph $$G$$ if and only if the maximum common edge subgraph of $$G$$ and $$H$$ has the same number of edges as $$H$$. Unless the two inputs $$G$$ and $$G'$$ to the maximum common edge subgraph problem are required to have the same number of vertices, the problem is APX-hard.