The Kemeny method is one of the popular tools for rank aggregation. However, computing an optimal Kemeny ranking is 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 first present a comprehensive relationship, both theoretical and empirical, among these parameters. Further, we study the problem of computing all distinct Kemeny rankings under the lens of parameterized complexity. 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 FPT algorithms. We find that any desirable number of Kemeny rankings can also be found without substantial increase in running time. We also present FPT approximation algorithms for Kemeny rank aggregation with respect to these parameters.

翻译：Kemeny方法是排序聚合的常用工具之一，然而计算最优Kemeny排序是NP难的。因此，寻找Kemeny排序的计算任务已在参数化复杂性视角下得到广泛研究，涉及多个参数。我们首先系统揭示了这些参数之间的理论关系与实证关系。进一步，我们研究了在参数化复杂性框架下计算所有不同Kemeny排序的问题。考虑的目标参数包括：目标Kemeny分数、候选者数量、输入排序的平均距离、任意候选者的最大范围以及一致宽度。针对这些参数，现有算法已实现固定参数可处理性（FPT）。我们发现，在无需显著增加运行时间的前提下，仍可得到任意所需数量的Kemeny排序。我们还针对这些参数提出了Kemeny排序聚合的FPT近似算法。