发表机构
Boston University（波士顿大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
针对批准委员会选举，提出在候选不相交批准链上精确计算按规模最小核心值的方法，证明其与活跃分量数的依赖关系，并验证余量的 Lipschitz 性质，从而保证核心非空性。
AI 中文摘要
对于基于批准的委员会选举，按规模最小核心值 $\Gamma(P)$ 最小化偏离规模为 $k$ 的委员会的最大超额支持。偏离是非空的且包含至多 $k$ 个候选人，其支持度相对于其 Hare 配额进行衡量。负值证明核心非空，而 $-\Gamma(P)$ 是任何委员会保证的最大最坏情况配额缺口。我们在候选不相交批准链上量化这一余量，该领域的定性核心非空性已知。我们首先将每个委员会的偏离分数分解到不相交的组成部分上。对于链，一个封闭的局部表格在多项式时间内为有理权重提供精确值和达到该值的委员会。对于非负归一化实权重，令 $q_{\mathrm{act}}\ge1$ 为正支持关联图的连通分量数。则 \\[ \Gamma(P)\le-\frac{1}{k\min\{q_{\mathrm{act}},k+1\}}. \\] 显式的极值分布表明这种对活跃分量数的依赖是精确的。该余量在完整选票空间上关于全变差是 1-Lipschitz 的。因此，每个负余量都证明了一个具有非空核心的邻域，即使扰动离开链域。
英文摘要
For approval-based committee elections, the sizewise least-core value $Γ(P)$ minimizes the maximum excess support of a deviation over size-$k$ committees. Deviations are nonempty and contain at most $k$ candidates, with support measured relative to their Hare quota. A negative value certifies core nonemptiness, while $-Γ(P)$ is the largest worst-case quota shortfall secured by any committee. We quantify this margin on candidate-disjoint approval chains, a domain whose qualitative core nonemptiness is already known. We first factor each committee's deviation score over disjoint components. For chains, a closed local table yields the exact value and an attaining committee in polynomial time for rational weights. For nonnegative normalized real weights, let $q_{\mathrm{act}}\ge1$ count the connected components of the positive-support incidence graph. Then \[ Γ(P)\le-\frac{1}{k\min\{q_{\mathrm{act}},k+1\}}. \] Explicit extremal profiles show that this dependence on the active-component count is sharp. The margin is $1$-Lipschitz in total variation on the full ballot space. Each negative margin therefore certifies a neighborhood with nonempty core, even when perturbations leave the chain domain.
Comments14 pages, code and other materials are available at: https://github.com/Baymax-ray/sizewise-least-core
