Summary
Picking groups that fairly represent voters and also satisfy efficiency is important but hard to do quickly. The authors study a popular but computationally tough voting rule called proportional approval voting (PAV) and show that a simpler version called local PAV nearly meets fairness and efficiency but has clear limits. They then create a new fast algorithm that combines the strengths of both approaches to produce committees that balance fairness, efficiency, and score quality well. This new method matches the best scores possible in reasonable time, assuming certain complexity limits. Their work helps us understand how to select good committees without impossible computations.
approval votingproportional approval votingcommittee selectionPareto optimalitylocal searchapproximation algorithmpolynomial timepipage roundingcomputational complexityEJR+
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 $α^\star$-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.