Multivariate Analyis of Swap Bribery

TL;DR

This paper analyzes the computational complexity of Swap Bribery in voting systems, exploring how different parameters and cost functions affect problem difficulty, and identifying cases of polynomial-time solvability versus NP-hardness.

Contribution

It provides a multivariate and parameterized complexity analysis of Swap Bribery, including polynomial-time algorithms and hardness results for various scenarios.

Findings

Polynomial-time solvable case for a special k-approval scenario

NP-hardness for a more general k-approval case

Fixed-parameter tractability and W[1]-hardness results for natural parameters

Abstract

We consider the computational complexity of a problem modeling bribery in the context of voting systems. In the scenario of Swap Bribery, each voter assigns a certain price for swapping the positions of two consecutive candidates in his preference ranking. The question is whether it is possible, without exceeding a given budget, to bribe the voters in a way that the preferred candidate wins in the election. We initiate a parameterized and multivariate complexity analysis of Swap Bribery, focusing on the case of k-approval. We investigate how different cost functions affect the computational complexity of the problem. We identify a special case of k-approval for which the problem can be solved in polynomial time, whereas we prove NP-hardness for a slightly more general scenario. We obtain fixed-parameter tractability as well as W[1]-hardness results for certain natural parameters.