Parameterized aspects of distinct Kemeny rank aggregation

The Kemeny method is one of the popular tools for rank aggregation. However, computing an optimal Kemeny ranking is \(\textsf{NP}\)-hard. Consequently, the computational task of finding a Kemeny ranking has been studied under the lens of parameterized complexity with respect to many parameters. We study the parameterized complexity of the problem of computing all distinct Kemeny rankings. We consider the target Kemeny score, number of candidates, average distance of input rankings, maximum range of any candidate, and unanimity width as our parameters. For all these parameters, we already have \(\textsf{FPT}\) algorithms. We find that any desirable number of Kemeny rankings can also be found without substantial increase in running time. We also present \(\textsf{FPT}\) approximation algorithms for Kemeny rank aggregation with respect to these parameters.