On the Complexity of Destructive Bribery in Approval-Based Multi-winner Voting

TL;DR

This paper explores the complexity of several destructive bribery problems under five prestigious approval-based multi-winner voting rules, and for NP-hard problems the authors study their parameterized complexity with respect to meaningful parameters.

Abstract

A variety of constructive manipulation, control, and bribery problems for approval-based multiwinner voting have been extensively studied recently. However, their destructive counterparts seem to be less explored. This paper investigates the complexity of several destructive bribery problems under five prestigious approval-based multiwinner voting rules -- approval voting, satisfaction approval voting, net-satisfaction approval voting, Chamberlin-Courant approval voting, and proportional approval voting. Broadly, these problems are to determine if a number of given candidates can be excluded from any winning committees by performing a limited number of modification operations. We offer a complete landscape of the complexity of the problems. For NP-hard problems, we study their parameterized complexity with respect to meaningful parameters.

Figures (2)

• Figure 1: An illustration of the reduction presented in the proof of Theorem \ref{['thm-appdel-sav-nsav-wah-l-k']}. For clarity, only the vertices in two groups, $U_i$ and $U_j$, are depicted. Each edge in $G$ is either within the same group, such as the edge between $w$ and $u$, or spans two different groups, like the edge between $u$ and $u'$. For every edge, there is a corresponding vote that approves exactly its two endpoints along with the distinguished candidate $p$. To simplify the illustration, only the votes corresponding to the edges $\{w,u\}$ and $\{u,u'\}$ are shown.
• Figure 2: An illustration of the election construction in the proof of Theorem \ref{['thm-pvc-sav-nsav-wah-l-k-r-1']}. Each set $C(u, i)$ consists of exactly $r+1$ candidates, while each set $C(\{u,u'\})$, created for an edge $\{u,u'\}$, consists of exactly $r-1$ candidates.

Theorems & Definitions (60)

• Theorem 1
• proof
• Theorem 2
• proof
• Corollary 1
• Lemma 1
• proof
• Theorem 3
• proof
• Theorem 4