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December 2021 The planted matching problem: Phase transitions and exact results
Mehrdad Moharrami, Cristopher Moore, Jiaming Xu
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Ann. Appl. Probab. 31(6): 2663-2720 (December 2021). DOI: 10.1214/20-AAP1660

Abstract

We study the problem of recovering a planted matching in randomly weighted complete bipartite graphs Kn,n. For some unknown perfect matching M, the weight of an edge is drawn from one distribution P if eM and another distribution Q if eM. Our goal is to infer M, exactly or approximately, from the edge weights. In this paper we take P=exp(λ) and Q=exp(1/n), in which case the maximum-likelihood estimator of M is the minimum-weight matching Mmin. We obtain precise results on the overlap between M and Mmin, that is, the fraction of edges they have in common. For λ4 we have almost perfect recovery, with overlap 1o(1) with high probability. For λ<4 the expected overlap is an explicit function α(λ)<1: we compute it by generalizing Aldous’ celebrated proof of the ζ(2) conjecture for the unplanted model, using local weak convergence to relate Kn,n to a type of weighted infinite tree, and then deriving a system of differential equations from a message-passing algorithm on this tree.

Funding Statement

The first author was supported by the Rackham Predoctoral Fellowship, a departmental Graduate Student Instructor appointment and by NSF Grant AST-1443972/AST-1516075. The majority of the work was done while the first author was at the University of Michigan.
The second author was supported in part by NSF Grant IIS-1838251.
The third author was supported by NSF Grants IIS-1838124, CCF-1850743, and CCF-1856424.

Acknowledgments

We are very grateful to Venkat Anantharam, Charles Bordenave, Jian Ding, David Gamarnik, Christopher Jones, Vijay Subramanian, Yihong Wu and Lenka Zdeborová for helpful conversations. C.M. is also grateful to Microsoft Research New England for their hospitality. We also thank an anonymous reviewer for helpful comments.

Citation

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Mehrdad Moharrami. Cristopher Moore. Jiaming Xu. "The planted matching problem: Phase transitions and exact results." Ann. Appl. Probab. 31 (6) 2663 - 2720, December 2021. https://doi.org/10.1214/20-AAP1660

Information

Received: 1 April 2020; Revised: 1 October 2020; Published: December 2021
First available in Project Euclid: 13 December 2021

Digital Object Identifier: 10.1214/20-AAP1660

Subjects:
Primary: 68Q87 , 90C27
Secondary: 05C70 , 05C80 , 62F15 , 82B26

Keywords: Combinatorial optimization , Local weak convergence , message-passing algorithms , Phase transitions , planted problems , Random graphs

Rights: Copyright © 2021 Institute of Mathematical Statistics

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Vol.31 • No. 6 • December 2021
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