The Planted Matching Problem: Phase Transitions and Exact Results

TL;DR

This paper analyzes the phase transition in recovering a planted matching in a weighted bipartite graph, providing exact thresholds and overlap formulas using probabilistic and message-passing techniques.

Contribution

It introduces precise phase transition results for the planted matching problem, extending Aldous' proof and employing local weak convergence and differential equations.

Findings

Almost perfect recovery for  ormat 4 with high probability

Explicit formula for expected overlap  < 4

Uses local weak convergence and message-passing on infinite trees

Abstract

We study the problem of recovering a planted matching in randomly weighted complete bipartite graphs $K_{n,n}$. For some unknown perfect matching $M^*$, the weight of an edge is drawn from one distribution $P$ if $e \in M^*$ and another distribution $Q$ if $e \notin M^*$. Our goal is to infer $M^*$, exactly or approximately, from the edge weights. In this paper we take $P=\exp(\lambda)$ and $Q=\exp(1/n)$, in which case the maximum-likelihood estimator of $M^*$ is the minimum-weight matching $M_{\text{min}}$. We obtain precise results on the overlap between $M^*$ and $M_{\text{min}}$, i.e., the fraction of edges they have in common. For $\lambda \ge 4$ we have almost perfect recovery, with overlap $1-o(1)$ with high probability. For $\lambda < 4$ the expected overlap is an explicit function $\alpha(\lambda) < 1$: we compute it by generalizing Aldous' celebrated proof of the $\zeta(2)$ conjecture for the un-planted model, using local weak convergence to relate $K_{n,n}$ to a type of weighted infinite tree, and then deriving a system of differential equations from a message-passing algorithm on this tree.