[Paper Review] Multistage Committee Election.

Robert Bredereck, Till Fluschnik+1 more|arXiv (Cornell University)|May 5, 2020

Game Theory and Voting Systems12 references4 citations

TL;DR

This paper introduces two time-dependent multistage models—conservative and revolutionary—for sequential Plurality-based committee elections, where consecutive committees must differ by a bounded symmetric difference. It proves both models NP-hard even with a constant number of agents, and shows that while the conservative model remains W[1]-hard and NP-hard, the revolutionary model becomes polynomial-time solvable when the symmetric difference bound is constant.

ABSTRACT

Electing a single committee of a small size is a classical and well-understood voting situation. Being interested in a sequence of committees, we introduce and study two time-dependent multistage models based on simple Plurality voting. Therein, we are given a sequence of voting profiles (stages) over the same set of agents and candidates, and our task is to find a small committee for each stage of high score. In the conservative model we additionally require that any two consecutive committees have a small symmetric difference. Analogously, in the revolutionary model we require large symmetric differences. We prove both models to be NP-hard even for a constant number of agents, and, based on this, initiate a parameterized complexity analysis for the most natural parameters and combinations thereof. Among other results, we prove both models to be in XP yet W[1]-hard regarding the number of stages, and that being revolutionary seems to be easier than being conservative: If the (upper- resp. lower-) bound on the size of symmetric differences is constant, the conservative model remains NP-hard while the revolutionary model becomes polynomial-time solvable.

Motivation & Objective

• To model sequential committee elections over time using Plurality voting with temporal constraints on committee changes.
• To formalize two contrasting models: conservative (small changes between consecutive committees) and revolutionary (large changes).
• To analyze the computational complexity of both models, especially under parameterized settings.
• To identify tractable cases and understand the relative difficulty of conservative versus revolutionary committee selection.

Proposed method

• Formalize a multistage voting model where each stage has a distinct voting profile over the same set of agents and candidates.
• Define the conservative model requiring that the symmetric difference between consecutive committees is at most a given upper bound.
• Define the revolutionary model requiring that the symmetric difference between consecutive committees is at least a given lower bound.
• Prove NP-hardness for both models even with a constant number of agents, using reductions from known NP-hard problems.
• Conduct a parameterized complexity analysis focusing on the number of stages and symmetric difference bounds.
• Establish that the conservative model is W[1]-hard when parameterized by the number of stages, while the revolutionary model becomes polynomial-time solvable under constant symmetric difference bounds.

Experimental results

Research questions

• RQ1 What is the computational complexity of the multistage conservative Plurality committee election problem?
• RQ2 How does the complexity of the multistage revolutionary Plurality committee election problem compare to the conservative variant?
• RQ3 Under which parameter combinations does the conservative model remain intractable or become tractable?
• RQ4 Can the revolutionary model be solved in polynomial time when the symmetric difference bound is constant?
• RQ5 Is there a significant complexity gap between conservative and revolutionary models under bounded symmetric difference constraints?

Key findings

• Both the conservative and revolutionary multistage Plurality committee election models are NP-hard, even when the number of agents is constant.
• The conservative model remains NP-hard and W[1]-hard when parameterized by the number of stages.
• The revolutionary model becomes polynomial-time solvable when the lower bound on symmetric differences is constant.
• The conservative model is in XP with respect to the number of stages, but not fixed-parameter tractable unless FPT = W[1].
• The revolutionary model is computationally easier than the conservative model under bounded symmetric difference constraints.
• The complexity gap between the two models is significant: bounded symmetric differences render the revolutionary model tractable while the conservative model remains hard.