Close and Sort: Exact Maximum-Likelihood Aggregation of Cyclic Rank-Two Preferences

Kehan Liu, Zhaoyun Chen, Xianquan Yan

Pairwise probabilities arise in rank aggregation, recommendation, and human or LLM evaluation, yet converting them into a maximum-likelihood list is a weighted Kemeny problem and is NP-hard in general. Known exact algorithms for rank-two tournaments use only comparison signs and do not optimize this likelihood. We show that the weighted problem is exactly solvable when the skew-symmetric log-odds matrix $W_{ij}=\log(p_{ij}/p_{ji})$ has rank two, including when the induced preferences contain cycles. Writing $W_{ij}=\det(h_i,h_j)$ and adding the closing vector $h_0=-\sum_i h_i$ turns the objective into twice the signed area of a closed polygon. This geometry yields CloseSort, which polar-sorts the $n+1$ edge vectors, cuts after $h_0$, and returns a global maximum-likelihood order in $O(n\log n)$ time from the factors. The same geometry identifies the runner-up and gives a stability certificate beyond exact rank two. When the smallest determinant $δ$ of two consecutive edges in the closed angular cycle is positive, it is exactly the best-to-runner-up log-likelihood gap. For $W=W_0+E$, the CloseSort order of $W_0$ has regret at most $\sum_{i<j}|E_{ij}|$ on $W$ and remains optimal…