We study the fair division of indivisible items among $n$ agents with heterogeneous additive valuations, subject to lower and upper quotas on the number of items allocated to each agent. Such constraints are crucial in various applications, ranging from personnel assignments to computing resource distribution. This paper focuses on the fairness criterion known as maximin shares (MMS) and its approximations. Under arbitrary lower and upper quotas, we show that a $\left(\frac{2n}{3n-1}\right)$-MMS allocation of goods exists and can be computed in polynomial time, while we also present a polynomial-time algorithm for finding a $\left(\frac{3n-1}{2n}\right)$-MMS allocation of chores. Furthermore, we consider the generalized scenario where items are partitioned into multiple categories, each with its own lower and upper quotas. In this setting, our algorithm computes an $\left(\frac{n}{2n-1}\right)$-MMS allocation of goods or a $\left(\frac{2n-1}{n}\right)$-MMS allocation of chores in polynomial time. These results extend previous work on the cardinality constraints, i.e., the special case where only upper quotas are imposed.

翻译：本文研究在异质加性估值下，将不可分割物品分配给$n$个智能体的问题，其中每个智能体分配到的物品数量受到上下界约束。此类约束在人员配置、计算资源分配等多种应用场景中至关重要。本文聚焦于最大最小份额（MMS）及其近似这一公平性准则。在任意上下界约束下，我们证明存在一种$\left(\frac{2n}{3n-1}\right)$-MMS的商品分配方案，并可在多项式时间内计算得到；同时我们也提出一种多项式时间算法，用于寻找$\left(\frac{3n-1}{2n}\right)$-MMS的杂务分配方案。此外，我们考虑更一般的场景：物品被划分为多个类别，每个类别具有独立的上下界约束。在此设定下，我们的算法可在多项式时间内计算出$\left(\frac{n}{2n-1}\right)$-MMS的商品分配方案或$\left(\frac{2n-1}{n}\right)$-MMS的杂务分配方案。这些结果扩展了先前仅考虑基数约束（即仅施加上界约束的特殊情形）的研究工作。