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 √ dB(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 √ dB(P ,Q) ≥ 1 + ǫ for an arbitrarily small constant ǫ > 0, the reconstruction error for any estimator is shown to be bounded away from 0 under both the sparse and dense model, resolving the conjecture in [20, 24]. 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−ǫ, the optimal reconstruction error is exp (−Θ(1/√ǫ)), confirming the conjectured infinite-order phase transition in [24].