Finding Representative and Approximately Efficient Committees

2026-09-07

AI Summary

本文探讨了通过计算效率高的方法实现PAV的公正性和效率性，提出了一种结合全局近似保证和局部搜索稳定性的多项式时间算法。

Abstract

In approval-based committee voting, proportional approval voting (PAV) is a well-studied rule that combines proportional representation with Pareto efficiency. However, computing a PAV committee is NP-hard, raising a natural question: Can the proportionality and efficiency properties of PAV be achieved via computationally efficient procedures? We make two contributions toward answering this question. First, building on the known proportionality guarantees of the local-search-based variant of PAV (or local PAV), we systematically study its efficiency properties. We show that local PAV committees are weakly Pareto optimal, meaning that no other committee is strictly preferred by every voter. We also identify limitations: Local PAV guarantees only a $2$-approximation to fractional Pareto optimality ($2$-fPO) and a $2/3$-approximation to the optimal PAV score, and both bounds are tight. In contrast, global PAV is Pareto optimal and satisfies the stronger $\alpha^\star$-fPO guarantee, where $\alpha^\star \approx 1.346$ is the unique solution of $\int_0^{\alpha^\star} \frac{1-e^{-y}}{y} \, dy = 1$, and this approximation is tight. Second, we design a polynomial-time algorithm that combines the best of these guarantees. The committee returned by our algorithm satisfies EJR$+$ (a proportionality guarantee), $\alpha^\star$-fPO, and weak Pareto optimality. It also achieves a $0.79$-approximation to the optimal PAV score, matching the best possible polynomial-time approximation assuming $P \neq NP$. Our algorithm works by pipage rounding a concave relaxation of the PAV objective and using that committee to initialize local PAV, thereby combining global approximation guarantees with local search stability.

Problem

• proportional approval voting
• Pareto efficiency
• computational efficiency
• committee voting
• approximation

Innovation

• polynomial-time algorithm
• EJR+
• α*-fPO
• weak Pareto optimality
• pipage rounding