Improved Parameterized Algorithms for the Kemeny Aggregation Problem

Abstract

We give improvements over fixed parameter tractable (FPT) algo-rithms 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 or-dered differently in the two lists. We study the problem for preference lists that are total orders. We develop algorithms of running times O∗(1.403kt), O∗(5.823kt/m) ≤ O∗(5.823kavg) and O∗(4.829kmax) for the problem, ignoring the polynomial factors in the O ∗ notation, where kt is the optimum total τ-distance, m is the number of votes, and kavg (resp, kmax) is the average (resp, maximum) over pairwise τ-distances of votes. Our algorithms improve the best previously known running times of O∗(1.53kt) and O∗(16kavg) ≤ O∗(16kmax) [4, 5], which also implies an O∗(164kt/m) running time. We also show how to enumerate all optimal solutions in O∗(36kt/m) ≤ O∗(36kavg) time.