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 observation in mind, we take a first step beyond the fixed-quota paradigm and introduce proportionality notions where the quota is instance-dependent. We demonstrate that all commonly studied voting rules can have an additive distance to the optimum of $\frac{k^2}{(k+1)^2}$. Moreover, we look into the computational aspects of our instance-dependent quota and prove that determining the optimal value of $\alpha$ for a given approval profile satisfying $\alpha$-JR is NP-complete. To address this, we introduce an integer linear programming (ILP) formulation for computing committees that satisfy $\alpha$-JR, and we provide positive results in the voter interval (VI) and candidate interval (CI) domains.

翻译：在基于批准投票的多赢家选举中，合理性代表（JR）与扩展合理性代表（EJR）是已确立的比例性公理。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解的最小配额。基于这一观察，我们迈出了超越固定配额范式的第一步，引入了配额依赖于具体实例的比例性概念。我们证明，所有常见投票规则与最优解之间的加性距离可达 $\frac{k^2}{(k+1)^2}$。此外，我们研究了实例依赖配额的计算特性，并证明对于给定的批准投票概况，确定满足 $\alpha$-JR 的最优 $\alpha$ 值是 NP 完全问题。为此，我们提出了用于计算满足 $\alpha$-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）域中给出了积极结果。