Kemeny Consensus is a well-known rank aggregation method in social choice theory. In this method, given a set of rankings, the goal is to find a ranking $Π$ that minimizes the total Kendall tau distance to the input rankings. Computing a Kemeny consensus is NP-hard, and even verifying whether a given ranking is a Kemeny consensus is coNP-complete. Fitzsimmons and Hemaspaandra [IJCAI 2021] established the computational intractability of achieving a desired consensus through manipulative actions. Kemeny Consensus is an optimisation problem related to Kemeny's rule. In this paper, we consider a decision problem related to Kemeny's rule, known as Kemeny Score, in which the goal is to decide whether there exists a ranking $Π$ whose total Kendall tau distance from the given rankings is at most $k$. Computation of Kemeny score is known to be NP-complete. In this paper, we investigate the impact of several manipulation actions on the Kemeny Score problem, in which given a set of rankings, an integer $k$, and a ranking $X$, the question is to decide whether it is possible to manipulate the given rankings so that the total Kendall tau distance of $X$ from the manipulated rankings is at most $k$. We show that this problem can be solved in polynomial time for various manipulation actions. Interestingly, these same manipulation actions are known to be computationally hard for Kemeny consensus.

翻译：肯梅尼共识是社会选择理论中一种著名的排序聚合方法。该方法中，给定一组排序，目标是找到一个排序Π，使其与输入排序的总肯德尔tau距离最小化。计算肯梅尼共识是NP难的，甚至验证给定排序是否为肯梅尼共识也是coNP完全的。Fitzsimmons和Hemaspaandra [IJCAI 2021]证明了通过操纵行为实现期望共识的计算不可行性。肯梅尼共识是与肯梅尼规则相关的一个优化问题。本文中，我们考虑与肯梅尼规则相关的一个决策问题，称为肯梅尼得分，其目标是判断是否存在一个排序Π，使其与给定排序的总肯德尔tau距离最多为k。已知肯梅尼得分的计算是NP完全的。本文研究了多种操纵行为对肯梅尼得分问题的影响，其中，给定一组排序、一个整数k和一个排序X，问题在于判断是否可能操纵给定排序，使得X与操纵后排序的总肯德尔tau距离最多为k。我们证明，对于多种操纵行为，该问题可在多项式时间内求解。有趣的是，已知这些相同的操纵行为对于肯梅尼共识在计算上是困难的。