Noisy sorting without resampling

In this paper we study noisy sorting without re-sampling. In this problem there is an unknown order {display equation} where π is a permutation on <i>n</i> elements. The input is the status of (^<i>n</i>₂) queries of the form <i>q</i>(<i>a_i, a_j</i>), for <i>i</i> < <i>j</i>, where <i>q(a_i, a_j)</i> = + (-) with probability 1/2 + γ if π(<i>i</i>) > π(<i>j</i>)(π(<i>i</i>) < π(<i>j</i>)) for all pairs <i>i</i> ≠ <i>j</i>, where γ > 0 is a constant. It is assumed that the errors are independent. Given the status of the queries the goal is to find the maximum likelihood order. In other words, the goal is find a permutation σ that minimizes the number of pairs σ(<i>i</i>) > σ(<i>j</i>) where <i>q</i>(σ(<i>i</i>), σ(<i>j</i>)) = -. The problem so defined is the feedback arc set problem on distributions of inputs, each of which is a tournament obtained as a noisy perturbation of a linear order. Note that when γ < 1/2 and <i>n</i> is large, it is impossible to recover the original order π. It is known that the weighted feedback arc set problem on tournaments is NP-hard in general. Here we present an algorithm of running time <i>n^O(γ-4))</i> and sampling complexity <i>O</i>γ (<i>n</i> log <i>n</i>) that with high probability solves the noisy sorting without re-sampling problem. We also show that if <i>a</i>_σ(1), <i>a</i>_σ(2), …, <i>a_σ(n)</i> is an optimal solution of the problem then it is "close" to the original order. More formally, with high probability it holds that {display equation}. Our results are of interest in applications to ranking, such as ranking in sports, or ranking of search items based on comparisons by experts.