We study the problem of recovering a planted matching in randomly weighted complete bipartite graphs . For some unknown perfect matching , the weight of an edge is drawn from one distribution P if and another distribution Q if . Our goal is to infer , exactly or approximately, from the edge weights. In this paper we take and , in which case the maximum-likelihood estimator of is the minimum-weight matching . We obtain precise results on the overlap between and , that is, the fraction of edges they have in common. For we have almost perfect recovery, with overlap with high probability. For the expected overlap is an explicit function : we compute it by generalizing Aldous’ celebrated proof of the conjecture for the unplanted model, using local weak convergence to relate to a type of weighted infinite tree, and then deriving a system of differential equations from a message-passing algorithm on this tree.
The first author was supported by the Rackham Predoctoral Fellowship, a departmental Graduate Student Instructor appointment and by NSF Grant AST-1443972/AST-1516075. The majority of the work was done while the first author was at the University of Michigan.
The second author was supported in part by NSF Grant IIS-1838251.
The third author was supported by NSF Grants IIS-1838124, CCF-1850743, and CCF-1856424.
We are very grateful to Venkat Anantharam, Charles Bordenave, Jian Ding, David Gamarnik, Christopher Jones, Vijay Subramanian, Yihong Wu and Lenka Zdeborová for helpful conversations. C.M. is also grateful to Microsoft Research New England for their hospitality. We also thank an anonymous reviewer for helpful comments.
"The planted matching problem: Phase transitions and exact results." Ann. Appl. Probab. 31 (6) 2663 - 2720, December 2021. https://doi.org/10.1214/20-AAP1660