A polynomial-time approximation scheme for the maximal overlap of two independent Erd\H{o}s-R\'enyi graphs

TL;DR

This paper introduces a polynomial-time approximation scheme for maximizing the overlap of two independent Erd ext{o}s-R ext{e}nyi graphs, providing near-optimal solutions and asymptotic analysis for specific edge probabilities.

Contribution

The paper develops a polynomial-time algorithm that approximates the maximum overlap of two Erd ext{o}s-R ext{e}nyi graphs arbitrarily closely, and derives asymptotic formulas for the maximum overlap.

Findings

Algorithm achieves near-optimal overlap approximation.

Asymptotic maximum overlap is $rac{n}{2eta-1}$ for $p=n^{-eta}$.

Provides theoretical bounds and analysis for graph overlap.

Abstract

For two independent Erd\H{o}s-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 $\frac{n}{2\alpha-1}$ for $p=n^{-\alpha}$ with some constant $\alpha\in (1/2,1)$.