election software developers

Abstract: PJR$^+$ is a polynomial-time verifiable proportionality axiom for approval-based committee elections, but its known polynomial-time verification procedure relies on general submodular-function minimisation. We show that its objective is a maximum-closure problem and give a direct mincut formulation of the problem on a bipartite graph. Using an almost-linear-time maximum-flow algorithm, this yields an $\mathcal{O}(m(nk)^{1+o(1)})$-time verifier, where $n$, $m$, and $k$ are the numbers of voters, candidates, and committee members, respectively. The dependence of this bound on each parameter separately is almost linear: it is linear in $m$, and almost linear in $n$ and $k$. The verifier also returns an explicit group witnessing a violation and admits a slower but immediately implementable variant based on the preflow--push mincut algorithm. Finally, for the parameterised axiom $α$-PJR$^+$, where $α$ is used as a multiplier in the group size, we demonstrate how to compute the largest value of $α$ for which a committee still fails the axiom using this mincut formulation.

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.