Abstract
We study the problem of recovering a planted matching in randomly weighted complete bipartite graphs Kn,n. For some unknown perfect matching Mâ, the weight of an edge is drawn from one distribution P if eâMâ and another distribution Q if eâMâ. Our goal is to infer Mâ, 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 Mâ is the minimum-weight matching Mmin. We obtain precise results on the overlap between Mâ and Mmin, that is, the fraction of edges they have in common. For λâ„4 we have almost perfect recovery, with overlap 1âo(1) with high probability. For λ<4 the expected overlap is an explicit function α(λ)<1: we compute it by generalizing Aldousâ 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.
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