The Planted Matching Problem: Phase Transitions and Exact Results
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 maximumlikelihood estimator of $M^*$ is the minimumweight 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 $1o(1)$ 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 unplanted 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 messagepassing algorithm on this tree.
 Publication:

arXiv eprints
 Pub Date:
 December 2019
 arXiv:
 arXiv:1912.08880
 Bibcode:
 2019arXiv191208880M
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Condensed Matter  Statistical Mechanics;
 Mathematics  Combinatorics