Improved Parameterized Algorithms for the Kemeny Aggregation Problem

Abstract

We give improvements over fixed parameter tractable (FPT) algorithms to solve the Kemeny aggregation problem, where the task is to summarize a multi-set of preference lists, called votes, over a set of alternatives, called candidates, into a single preference list that has the minimum total τ-distance from the votes. The τ-distance between two preference lists is the number of pairs of candidates that are ordered differently in the two lists. We study the problem for preference lists that are total orders. We develop algorithms of running times $O^*(1.403^{k_t})$, $O^*(5.823^{k_t/m})\leq O^*(5.823^{k_{avg}})$ and $O^*(4.829^{k_{max}})$ for the problem, ignoring the polynomial factors in the O ^* notation, where k _t is the optimum total τ-distance, m is the number of votes, and k _avg (resp. k _max ) is the average (resp. maximum) over pairwise τ-distances of votes. Our algorithms improve the best previously known running times of $O^*(1.53^{k_t})$ and $O^*(16^{k_{avg}})\leq O^*(16^{k_{max}})$ [3,4], which also implies an $O^*(16^{2k_t/m})$ running time. We also show how to enumerate all optimal solutions in $O^*(36^{k_t/m}) \leq O^*(36^{k_{avg}})$ time.