A polynomial‐time approximation scheme for the maximal overlap of two independent Erdős–Rényi graphs - Ding, Jian; Du, Hang; Gong, Shuyang (Random structures & algorithms, 2024)
For two independent Erdős–Rényi graphs G(n,p)$$ G\left(n,p\right) $$, 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 n2α−1$$ \frac{n}{2α -1} $$ for p=n−α$$ p={n}^{-α } $$ with some constant α∈(1/2,1)$$ α \in \left(1/2,1\right) $$.