Justified representation (JR) and extended justified representation (EJR) are well-established proportionality axioms in approval-based multiwinner voting. Both axioms are always satisfiable, but they rely on a fixed quota (typically Hare or Droop), with the Droop quota being the smallest one that guarantees existence across all instances. With this in mind, we take a step beyond the fixed-quota paradigm by studying instance-dependent proportionality notions. More specifically, we minimize the quota requirements for JR and EJR using the parameter $α$. We demonstrate that all commonly studied voting rules can have an additive gap to the optimum of $\frac{k^2}{(k+1)^2}$. Moreover, we examine the computational aspects of our instance-dependent quota and prove that determining the optimal value of $α$ for a given approval profile that allows some committee to satisfy $α$-JR is NP-complete. To address this, we introduce an integer linear programming (ILP) formulation for computing committees that satisfy $α$-JR, and we provide positive computational results in the voter interval (VI) and candidate interval (CI) domains.

翻译：在基于批准制的多赢家投票中，合理代表（JR）与扩展合理代表（EJR）是公认的比例性公理。这两条公理始终可满足，但它们依赖于固定配额（通常为黑尔配额或德鲁普配额），其中德鲁普配额是保证在所有实例中均存在满足条件委员会的最小配额。基于此，我们超越固定配额范式，研究依赖于具体实例的比例性概念。具体而言，我们利用参数 $α$ 最小化 JR 与 EJR 的配额要求。我们证明，所有常见投票规则与最优值之间可能存在 $\frac{k^2}{(k+1)^2}$ 的加性差距。此外，我们研究了实例依赖配额的计算复杂性，并证明对于给定的批准偏好剖面，确定使某个委员会满足 $α$-JR 的最优 $α$ 值是 NP 完全问题。为此，我们提出了用于计算满足 $α$-JR 的委员会的整数线性规划（ILP）模型，并在选民区间（VI）与候选人区间（CI）领域给出了积极的计算结果。