
The planted matching problem: Sharp threshold and infiniteorder phase transition
We study the problem of reconstructing a perfect matching M^* hidden in ...
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Revisiting the Auction Algorithm for Weighted Bipartite Perfect Matchings
We study the classical weighted perfect matchings problem for bipartite ...
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A PseudoDeterministic RNC Algorithm for General Graph Perfect Matching
The difficulty of obtaining an NC perfect matching algorithm has led res...
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Recovery thresholds in the sparse planted matching problem
We consider the statistical inference problem of recovering an unknown p...
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A new generalization of edge overlap to weighted networks
Finding the strength of an edge in a network has always been a big deman...
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Strong recovery of geometric planted matchings
We study the problem of efficiently recovering the matching between an u...
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Matching is as Easy as the Decision Problem, in the NC Model
We give an NC reduction from search to decision for the problem of findi...
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The Planted Matching Problem: Phase Transitions and Exact Results
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 maximumlikelihood estimator of M^* is the minimumweight 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 almostperfect recovery, with overlap 1o(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 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.
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