The Planted Matching Problem: Sharp Threshold and Infinite-Order Phase Transition

We study the problem of reconstructing a perfect matching M^* hidden in a randomly weighted n× n bipartite graph. The edge set includes every node pair in M^* and each of the n(n-1) node pairs not in M^* independently with probability d / n . The weight of each edge e is independently drawn from the distribution 𝒫 if e ∈ M^* and from 𝒬 if e ∉ M^* . We show that if √(d) B(𝒫,𝒬) ≤ 1 , where B(𝒫,𝒬) stands for the Bhattacharyya coefficient, the reconstruction error (average fraction of misclassified edges) of the maximum likelihood estimator of M^* converges to 0 as n→∞ . Conversely, if √(d) B(𝒫,𝒬) ≥ 1+ϵ for an arbitrarily small constant ϵ >0 , the reconstruction error for any estimator is shown to be bounded away from 0 for both the sparse (fixed d ) and dense (growing d ) regimes, resolving the conjecture in Moharrami et al. (Ann Appl Probab 31(6):2663–2720, 2021. https://doi.org/10.1214/20-AAP1660 ) and Semerjian et al. (Phys Rev E 102:022304, 2020. https://doi.org/10.1103/PhysRevE.102.022304 ). Furthermore, in the special case of complete exponentially weighted graph with d=n , 𝒫=exp (λ ) , and 𝒬=exp (1/n) , for which the sharp threshold simplifies to λ =4 , we prove that when λ = 4-ϵ , the optimal reconstruction error is exp( - (1/√(ϵ)) ) , confirming the conjectured infinite-order phase transition in Semerjian et al. (2020).