Algorithms for Destructive Shift Bribery

TL;DR

This paper analyzes the computational complexity of Destructive Shift Bribery in elections, identifying which voting rules allow polynomial solutions and which are NP-hard, with additional parameterized complexity results and efficient algorithms for specific cases.

Contribution

It provides a comprehensive complexity classification of Destructive Shift Bribery across various voting rules, contrasting with the constructive case, and introduces parameterized algorithms and hardness results.

Findings

Polynomial-time solvable for scoring protocols, Bucklin, Simplified Bucklin, Maximin.

NP-hard for Copeland rule.

W-hardness results for certain parameterizations.

Abstract

We study the complexity of Destructive Shift Bribery. In this problem, we are given an election with a set of candidates and a set of voters (each ranking the candidates from the best to the worst), a despised candidate $d$, a budget $B$, and prices for shifting $d$ back in the voters' rankings. The goal is to ensure that $d$ is not a winner of the election. We show that this problem is polynomial-time solvable for scoring protocols (encoded in unary), the Bucklin and Simplified Bucklin rules, and the Maximin rule, but is NP-hard for the Copeland rule. This stands in contrast to the results for the constructive setting (known from the literature), for which the problem is polynomial-time solvable for $k$-Approval family of rules, but is NP-hard for the Borda, Copeland, and Maximin rules. We complement the analysis of the Copeland rule showing W-hardness for the parameterization by the budget value, and by the number of affected voters. We prove that the problem is W-hard when parameterized by the number of voters even for unit prices. From the positive perspective we provide an efficient algorithm for solving the problem parameterized by the combined parameter the number of candidates and the maximum bribery price (alternatively the number of different bribery prices).