“Noisy Sorting Without Resampling”, Mark Braverman, Elchanan Mossel2007-07-06 (sorting, statistical comparison; backlinks; similar)⁠:

In this paper we study noisy sorting without re-sampling. In this problem there is an unknown order a_π(1) <…< a_π(n) where π is a permutation on n elements. The input is the status of (ⁿ₂) queries of the form q(a_i, x_j), where q(a_i, a_j) = + with probability at least 1⁄2 +γ if π(i) > π(j) for all pairs i ≠ j, where γ > 0 is a constant and q(a_i, a_j) = −q(a_j, a_i) 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 σ that minimizes the number of pairs σ(i) > σ(j) where q(σ(i), σ(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 γ < 1⁄2 and n 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 n^{𝒪 (γ−4)}> and sampling complexity O_γ(n log n) that with high probability solves the noisy sorting without re-sampling problem. We also show that if a_σ(1), a_σ(2), …, a_σ(n) is an optimal solution of the problem then it is “close” to the original order. More formally, with high probability it holds that ∑_i|σ(i) − π(i)| = Θ(n) and max_i|σ(i) − π(i)| = Θ(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.

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