###### Cristopher Moore

What happens when an optimization problem has a good solution built into it, but which is partly obscured by randomness? Here we revisit a classic polynomial-time problem, the minimum perfect matching problem on bipartite graphs. If the edges have random weights in [0,1], Mézard and Parisi — and then Aldous, rigorously — showed that the minimum matching has expected weight zeta(2) = pi^2/6. We consider a “planted” version where a particular matching has weights drawn from an exponential distribution with mean mu/n. When mu < 1/4, the minimum matching is almost identical to the planted one. When mu > 1/4, the overlap between the two is given by a system of differential equations that result from a message-passing algorithm. This is joint work with Mehrdad Moharrami (Michigan) and Jiaming Xu (Duke).

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