Bribery Can Get Harder in Structured Multiwinner Approval Election

TL;DR

The paper investigates the complexity of constructive bribery in structured, multiwinner approval elections under candidate-interval (CI) and voter-interval (VI) preferences, allowing AddApprovals, DelApprovals, and SwapApprovals with costs. It provides a near-complete map of tractability versus hardness across CI/VI and pricing variants, revealing both polynomial-time algorithms (notably for AddApprovals and Swap-to-p cases) and NP-hardness (notably for several DelApprovals, Arbitrary Swaps, and destructive variants). The results are established through constructive algorithms and reductions from RX3C and Cubic Independent Set, illustrating interesting complexity reversals in structured domains. The practical impact lies in understanding when small bribery actions can feasibly influence committee composition under real-world, structured political or organizational settings. The work also offers methodological tools—dynamic programming and shortest-path analyses for VI, and carefully structured reductions—for analyzing similar problems in other voting rules or structures.

Abstract

We study the complexity of constructive bribery in the context of structured multiwinner approval elections. Given such an election, we ask whether a certain candidate can join the winning committee by adding, deleting, or swapping approvals, where each such action comes at a cost and we are limited by a budget. We assume our elections to either have the candidate interval or the voter interval property, and we require the property to hold also after the bribery. While structured elections usually make manipulative attacks significantly easier, our work also shows examples of the opposite behavior. We conclude by presenting preliminary insights regarding the destructive variant of our problem.

Figures (1)

• Figure 1: Illustration of the election from the proof of Theorem \ref{['thm:vi-swaps']}, for the case where $\boldsymbol{S_i = \{x_a,x_b,x_c\}}$, where $\boldsymbol{a = 1 < b <c}$. Each row corresponds to a voter and each column corresponds to a candidate. Solid boxes show approvals prior bribery and dotted ones show approval moves.

Theorems & Definitions (27)

• Example 1
• Example 2
• Definition 1
• Definition 2
• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof