How similarity helps to efficiently compute Kemeny rankings

The computation of Kemeny rankings is central to many applications in the context of rank aggregation. Unfortunately, the problem is NP-hard. We show that the Kemeny score (and a corresponding Kemeny ranking) of an election can be computed efficiently whenever the average pairwise distance between two input votes is not too large. In other words, Kemeny Score is fixed-parameter tractable with respect to the parameter “average pairwise Kendall-Tau distance da”. We describe a fixed-parameter algorithm with running time 16 da · poly. Moreover, we extend our studies to the parameters “maximum range” and “average range” of positions a candidate takes in the input votes. Whereas Kemeny Score remains fixed-parameter tractable with respect to the parameter “maximum range”, it becomes NPcomplete in case of an average range value of two. This excludes fixed-parameter tractability with respect to the parameter “average range” unless P=NP.