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 notin 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, i.e., 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 zeta(2) conjecture for the un-planted 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.