The planted matching problem: Phase transitions and exact results

Abstract

We study the problem of recovering a planted matching in randomly weighted complete bipartite graphs ${\mathit{K}}_{\mathit{n},\mathit{n}}$. For some unknown perfect matching ${\mathit{M}}^{\ast }$, the weight of an edge is drawn from one distribution P if $\mathit{e}\in {\mathit{M}}^{\ast }$ and another distribution Q if $\mathit{e}\notin {\mathit{M}}^{\ast }$. Our goal is to infer ${\mathit{M}}^{\ast }$, exactly or approximately, from the edge weights. In this paper we take $\mathit{P}=exp(\mathit{\lambda })$ and $\mathit{Q}=exp(1/\mathit{n})$, in which case the maximum-likelihood estimator of ${\mathit{M}}^{\ast }$ is the minimum-weight matching ${\mathit{M}}_{\min }$. We obtain precise results on the overlap between ${\mathit{M}}^{\ast }$ and ${\mathit{M}}_{\min }$, that is, the fraction of edges they have in common. For $\mathit{\lambda }\ge 4$ we have almost perfect recovery, with overlap $1-\mathit{o}(1)$ with high probability. For $\mathit{\lambda }<4$ the expected overlap is an explicit function $\mathit{\alpha }(\mathit{\lambda })<1$: we compute it by generalizing Aldous’ celebrated proof of the $\mathit{\zeta }(2)$ conjecture for the unplanted model, using local weak convergence to relate ${\mathit{K}}_{\mathit{n},\mathit{n}}$ to a type of weighted infinite tree, and then deriving a system of differential equations from a message-passing algorithm on this tree.