A polynomial-time approximation scheme for the maximal overlap of two independent Erd?s-Rnyi graphs

For two independent Erdos-Renyi graphs G(n,p)$$ \mathbf{G}\left(n,p\right) $$, 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 n2 alpha-1$$ \frac{n}{2\alpha -1} $$ for p=n-alpha$$ p={n}<^>{-\alpha } $$ with some constant alpha is an element of(1/2,1)$$ \alpha \in \left(1/2,1\right) $$.