🤖 AI Summary
This study addresses the problem of ranking and rank aggregation under matroid and flag-matroid prefix constraints, measured by Kendall tau distance. It unifies and generalizes existing notions such as k-fairness and block fairness, and for the first time handles more general constraints involving hierarchies and quotas. For the single-input setting, the authors propose a polynomial-time algorithm based on the Bruhat order and a greedy strategy to efficiently compute the nearest feasible ranking satisfying flag-matroid constraints. In the multi-input aggregation setting, they prove that the problem remains NP-hard even under partition matroids. By integrating matroid theory with structural analysis of the symmetric group, this work significantly extends the theoretical foundations of fair ranking.
📝 Abstract
We study ranking and rank aggregation under the Kendall tau distance, subject to matroid or flag matroid constraints on prefixes of the output ranking. In the matroid case, the top-$k$ prefix is required to form a base of a matroid; in the flag matroid case, several prescribed prefixes are required to form bases of a sequence of matroids linked by quotient relations. This framework contains the previously studied notions of $k$-fairness and block-fairness as special cases, and also captures more general hierarchical and assignment-type lower- and upper-quota constraints.
We provide a polynomial-time algorithm for finding, given a single input ranking, a closest feasible ranking under flag matroid prefix constraints. The algorithm is a natural greedy procedure, and its optimality is proved via a Bruhat order argument on the symmetric group. As a consequence, existing approximation frameworks for fair rank aggregation carry over to the matroidal setting. We also prove that rank aggregation with matroid constraints is NP-hard for every fixed number $m\ge 2$ of input rankings, even under partition matroid constraints.