The planted matching problem: Phase transitions and exact results

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 e∈M∗ and another distribution Q if e∉M∗. 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 λ≥4 we have almost perfect recovery, with overlap 1−o(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.