The Planted Matching Problem: Phase Transitions and Exact Results
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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 ∈ 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=(λ) and Q=(1/n), in which case the maximum-likelihood estimator of M^* is the minimum-weight matching M_min. We obtain precise results on the overlap between M^* and M_min, 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 Mézard and Parisi's ζ(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.
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