[Paper Review] On the Complexity of Destructive Bribery in Approval-Based Multi-winner Voting

Yongjie Yang|arXiv (Cornell University)|Feb 3, 2020

Game Theory and Voting Systems39 references4 citations

TL;DR

This paper investigates the computational complexity of destructive bribery in approval-based multi-winner voting systems, where an attacker seeks to exclude a set of distinguished candidates from all winning committees by modifying votes within a budget. It provides a complete complexity landscape across five prominent approval-based multi-winner rules (AV, SAV, NSAV, CCAV, PAV), proving NP-hardness for most problems and establishing fixed-parameter tractability (FPT) or W[1]-hardness with respect to key parameters like number of votes and budget.

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.

Motivation & Objective

• To address the lack of research on destructive bribery in approval-based multi-winner voting, which contrasts with extensive work on constructive variants.
• To model scenarios where an attacker aims to exclude multiple distinguished candidates from winning committees by modifying votes within a budget.
• To analyze the complexity of five bribery operations—approval addition, deletion, and three vote-level operations—under five prominent approval-based multi-winner voting rules.
• To provide a comprehensive parameterized complexity analysis, identifying FPT and W[1]-hard cases with respect to meaningful parameters such as number of votes, budget, and distance bounds.
• To offer a complete complexity landscape, including polynomial-time algorithms for special cases and NP-hardness results even in restricted settings.

Proposed method

• The paper models destructive bribery as a decision problem: can a set of distinguished candidates be excluded from all winning k-committees via at most ℓ modifications?
• It classifies operations into atomic (AppAdd, AppDel) and vote-level (VC, VAC, VDC), with VAC and VDC including a Hamming-distance bound r to limit deviation from original votes.
• For AV under AppAdd and AppDel, it uses a maximum network flow reduction to solve subinstances in polynomial time after enumerating candidate vote sets.
• For vote-level operations with r = 2m (maximum possible deviation), it uses enumeration over at most ℓ votes to be modified, followed by greedy assignment to exclude all distinguished candidates.
• It proves FPT results via enumeration and flow-based reductions, and establishes W[1]-hardness for certain parameter combinations.
• The analysis includes both classical complexity (NP-hardness) and parameterized complexity (FPT, W[1]-hardness) across five voting rules.

Experimental results

Research questions

• RQ1 What is the computational complexity of destructive bribery under approval voting (AV), satisfaction approval voting (SAV), net-satisfaction approval voting (NSAV), Chamberlin-Courant approval voting (CCAV), and proportional approval voting (PAV)?
• RQ2 How does the inclusion of a Hamming-distance bound r affect the complexity of vote-level bribery operations?
• RQ3 Which parameters—such as number of votes, budget, number of distinguished candidates, or committee size—render the problems fixed-parameter tractable or W[1]-hard?
• RQ4 Are there special cases where destructive bribery becomes polynomial-time solvable despite being NP-hard in general?
• RQ5 How do the results compare to known complexity results in constructive bribery and manipulation problems?

Key findings

• The destructive bribery problems under AV, SAV, NSAV, CCAV, and PAV are NP-hard, with many hardness results holding even in highly restricted cases.
• For AV with approval addition or deletion, the problem is solvable in polynomial time via maximum network flow reduction after enumerating candidate vote sets.
• For vote-level operations under AV with r = 2m, the problem is FPT with respect to the number of votes, solvable in O*(2^n) time.
• The parameterized complexity analysis reveals that while most problems are FPT with respect to the number of candidates, only a few are FPT with respect to the number of votes.
• Several problems are W[1]-hard when parameterized by the number of votes or budget, indicating strong intractability under these parameters.
• The study provides a complete complexity map across five voting rules and five operation types, filling a significant gap in the literature on destructive control and bribery.