A polynomial‐time approximation scheme for the maximal overlap of two independent Erdős–Rényi graphs

For two independent Erdős–Rényi 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 approxi...

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Bibliographic Details
Published in:Random structures & algorithms Vol. 65; no. 1; pp. 220 - 257
Main Authors: Ding, Jian, Du, Hang, Gong, Shuyang
Format: Journal Article
Language:English
Published: New York John Wiley & Sons, Inc 01.08.2024
Wiley Subscription Services, Inc
Subjects:
Algorithms
Graph theory
Graphs
greedy algorithm
Polynomials
polynomial‐time approximation scheme
random graph matching
ISSN:1042-9832, 1098-2418
Online Access:Get full text
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Summary:For two independent Erdős–Rényi 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 n2α−1$$ \frac{n}{2\alpha -1} $$ for p=n−α$$ p={n}^{-\alpha } $$ with some constant α∈(1/2,1)$$ \alpha \in \left(1/2,1\right) $$.
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ISSN:1042-9832
1098-2418
DOI:10.1002/rsa.21212