The Planted Matching Problem
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).
Cristopher Moore received his B.A. in Physics, Mathematics, and Integrated Science from Northwestern University, and his Ph.D. in Physics from Cornell. From 2000 to 2012 he was a professor at the University of New Mexico, with joint appointments in Computer Science and Physics. Since 2012, Moore has been a resident professor at the Santa Fe Institute; he has also held visiting positions at École Normale Superieure, École Polytechnique, Université Paris 7, the Niels Bohr Institute, Northeastern University, and the University of Michigan. He has published over 150 papers at the boundary between physics and computer science, ranging from quantum computing, to phase transitions in NP-complete problems, to the theory of social networks and efficient algorithms for analyzing their structure. He is an elected Fellow of the American Physical Society, the American Mathematical Society, and the American Association for the Advancement of Science. With Stephan Mertens, he is the author of The Nature of Computation from Oxford University Press.
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