Noisy Sorting Without Resampling

In this paper we study noisy sorting without re-sampling. In this problem there is an unknown order $a{\pi(1)} < ... < a{\pi(n)}$ where $\pi$ is a permutation on $n$ elements. The input is the status of $n \choose 2$ queries of the form $q(ai,xj)$, where $q(ai,aj) = +$ with probability at least $1/2+\ga$ if $\pi(i) > \pi(j)$ for all pairs $i \neq j$, where $\ga > 0$ is a constant and $q(ai,aj) = -q(aj,ai)$ for all $i$ and $j$. 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 $\sigma$ that minimizes the number of pairs $\sigma(i) > \sigma(j)$ where $q(\sigma(i),\sigma(j)) = -$. The problem so defined is the feedback arc set problem on distributions of inputs, each of which is a tournament obtained as a noisy perturbations of a linear order. Note that when $\ga < 1/2$ and $n$ is large, it is impossible to recover the original order $\pi$. It is known that the weighted feedback are set problem on tournaments is NP-hard in general. Here we present an algorithm of running time $n^{{O(\gamma{-4})}$} and sampling complexity $O{\gamma}(n \log n)$ that with high probability solves the noisy sorting without re-sampling problem. We also show that if $a{\sigma(1)},a{\sigma(2)},...,a{\sigma(n)}$ is an optimal solution of the problem then it is ``close'' to the original order. More formally, with high probability it holds that $\sumi |\sigma(i) - \pi(i)| = \Theta(n)$ and $\maxi |\sigma(i) - \pi(i)| = \Theta(\log n)$. Our results are of interest in applications to ranking, such as ranking in sports, or ranking of search items based on comparisons by experts.