Parameterized Enumeration of (Locally-) Optimal Aggregations

Abstract

We present a parameterized enumeration algorithm for Kemeny Rank Aggregation, the problem of determining an optimal aggregation, a total order that is at minimum total τ-distance (k _t ) from the input multi-set of m total orders (votes) over a set of alternatives (candidates), where the τ-distance between two total orders is the number of pairs of candidates ordered differently. Our $O^*(4^{k_t\over m})$-time algorithm constitutes a significant improvement over the previous $O^*(36^{k_t\over m})$ upper bound.

The analysis of our algorithm relies on the notion of locally-optimal aggregations, total orders whose total τ-distances from the votes do not decrease by any single swap of two candidates adjacent in the ordering. As a consequence of our approach, we provide not only an upper bound of $4^{k_t\over m}$ on the number of optimal aggregations, but also the first parameterized bound, $4^{k_t\over m}$, on the number of locally-optimal aggregations, and demonstrate that it is tight. Furthermore, since our results rely on a known relation to Weighted Directed Feedback Arc Set, we obtain new results for this problem along the way.