Bribery's Influence on Ranked Aggregation

TL;DR

This paper explores how certain manipulation actions affect the computational complexity of the Kemeny Score problem, showing some can be solved efficiently despite related problems being NP-hard.

Contribution

It demonstrates that specific manipulation actions allow polynomial-time solutions for the Kemeny Score problem, contrasting with their hardness in the context of Kemeny consensus.

Findings

Some manipulation actions enable polynomial-time solutions for Kemeny Score.

Manipulation actions are computationally hard for Kemeny consensus but tractable for Kemeny Score.

The study highlights differences in complexity between related rank aggregation problems.

Abstract

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 $\Pi$ 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 $\Pi$ 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.