A polynomial-time approximation scheme for the maximal overlap of two independent ErdHos-R'enyi graphs

For two independent ErdHos-R'enyi graphs mathbf G(n,p), we study the maximal overlap (i.e., the number of common edges) of these two graphs over all possible vertex correspondence. We present a polynomial-time algorithm which finds a vertex correspondence whose overlap approximates the maximal overlap up to a multiplicative factor that is arbitrarily close to 1. As a by-product, we prove that the maximal overlap is asymptotically fracn2α-1 for p=n-α with some constant α∈ (1/2,1).