寻找具有代表性和近似效率的委员会

中文摘要 AI 辅助

在基于批准的委员会投票中，比例批准投票（PAV）是一种被广泛研究的规则，它结合了比例代表制与帕累托效率。然而，计算一个PAV委员会是NP困难的，这引发了一个自然的问题：PAV的比例性和效率性质能否通过计算高效的程序实现？我们为回答这个问题做出了两项贡献。首先，基于已知的局部搜索变体PAV（或局部PAV）的比例性保证，我们系统地研究了其效率性质。我们证明局部PAV委员会是弱帕累托最优的，这意味着没有其他委员会被每个选民严格偏好。我们也指出了局限性：局部PAV仅保证分数帕累托最优性的$2$-近似（$2$-fPO）和最优PAV分数的$2/3$-近似，且这两个界都是紧的。相比之下，全局PAV是帕累托最优的，并满足更强的$\alpha^\star$-fPO保证，其中$\alpha^\star \approx 1.346$是$\int_0^{\alpha^\star} \frac{1-e^{-y}}{y} \\, dy = 1$的唯一解，且该近似是紧的。其次，我们设计了一个多项式时间算法，结合了这些保证的最佳特性。我们的算法返回的委员会满足EJR$+$（一种比例性保证）、$\alpha^\star$-fPO和弱帕累托最优性。它还实现了最优PAV分数的$0.79$-近似，匹配了假设$P \neq NP$下最佳可能的多项式时间近似。我们的算法通过对PAV目标的凹松弛进行管道舍入，并使用该委员会初始化局部PAV，从而结合了全局近似保证与局部搜索稳定性。

英文摘要

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 $α^\star$-fPO guarantee, where $α^\star \approx 1.346$ is the unique solution of $\int_0^{α^\star} \frac{1-e^{-y}}{y} \, dy = 1$, 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), $α^\star$-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.

发表机构

• CNRS, LAMSADE, Université Paris Dauphine - PSL（法国国家科学研究中心，LAMSADE实验室，巴黎第九大学-PSL）
• Indian Institute of Technology Delhi（德里印度理工学院）

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