通过最小割加速PJR$^+$的验证
Faster Verification of PJR$^+$ via Mincuts
- King’s College London（伦敦国王学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文针对PJR$^+$比例性公理的验证问题，提出基于最大闭包和二分图最小割的几乎线性时间验证算法，并扩展至参数化$\alpha$-PJR$^+$的最大失败$\alpha$计算。
AI中文摘要：
PJR$^+$是批准制委员会选举中一个可在多项式时间内验证的比例性公理，但其已知的多项式时间验证过程依赖于一般的子模函数最小化。我们证明其目标是一个最大闭包问题，并给出该问题在二分图上的直接最小割表述。利用一种几乎线性时间的最大流算法，这产生了一个$\mathcal{O}(m(nk)^{1+o(1)})$时间的验证器，其中$n$、$m$和$k$分别表示选民、候选人和委员会成员的数量。该界对每个参数的依赖分别几乎是线性的：对$m$是线性的，对$n$和$k$几乎是线性的。该验证器还返回一个显式的违规见证组，并支持一个基于预流-推送最小割算法的较慢但可立即实现的变体。最后，对于参数化公理$\alpha$-PJR$^+$（其中$\alpha$用作组大小的乘数），我们演示了如何利用该最小割表述计算使委员会仍不满足该公理的最大$\alpha$值。
英文摘要：
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.
