Ranking and Rank Aggregation with Matroid Prefix Constraints

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.