Bribery as a Measure of Candidate Success: Complexity Results for Approval-Based Multiwinner Rules

TL;DR

This paper analyzes the computational complexity of bribery in approval-based multiwinner elections, revealing a diverse landscape of polynomial-time solvability, NP-hardness, and inapproximability results, with some problems becoming easier under certain restrictions.

Contribution

It provides a comprehensive complexity analysis of bribery problems across various approval-based multiwinner rules, including new results on parameterized complexity and approximation hardness.

Findings

Complexity varies from polynomial-time to NP-hardness depending on the rule and action.

Limiting bribery actions to the preferred candidate simplifies some problems.

Parameterized complexity results highlight the influence of voter and candidate counts.

Abstract

We study the problem of bribery in multiwinner elections, for the case where the voters cast approval ballots (i.e., sets of candidates they approve) and the bribery actions are limited to: adding an approval to a vote, deleting an approval from a vote, or moving an approval within a vote from one candidate to the other. We consider a number of approval-based multiwinner rules (AV, SAV, GAV, RAV, approval-based Chamberlin--Courant, and PAV). We find the landscape of complexity results quite rich, going from polynomial-time algorithms through NP-hardness with constant-factor approximations, to outright inapproximability. Moreover, in general, our problems tend to be easier when we limit out bribery actions on increasing the number of approvals of the candidate that we want to be in a winning committee (i.e., adding approvals only for this preferred candidate, or moving approvals only to him or her). We also study parameterized complexity of our problems, with a focus on parameterizations by the numbers of voters or candidates.