On reconstructing a hidden permutation

The Mallows model is a classical model for generating noisy perturbations of a hidden permutation, where the magnitude of the perturbations is determined by a single parameter. In this work we consider the following reconstruction problem: given several perturbations of a hidden permutation that are generated according to the Mallows model, each with its own parameter, how to recover the hidden permutation? When the parameters are approximately known and satisfy certain conditions, we obtain a simple algorithm for reconstructing the hidden permutation; we also show that these conditions are nearly inevitable for reconstruction. We then provide an algorithm to estimate the parameters themselves. En route we obtain a precise characterization of the swapping probability in the Mallows model.