We study the problem of fair allocation of a set of indivisible items among agents with additive valuations, under matroid constraints and two generalizations: $p$-extendible system and independence system constraints. The objective is to find fair and efficient allocations in which the subset of items assigned to every agent satisfies the given constraint. We focus on a common fairness notion of envy-freeness up to one item (EF1) and a well-known efficient (and fair) notion of the maximum Nash social welfare (Max-NSW). By using properties of matroids, we demonstrate that the Max-NSW allocation, implying Pareto optimality (PO), achieves a tight $1/2$-EF1 under matroid constraints. This result resolves an open question proposed in prior literature [26]. In particular, if agents have 2-valued ($\{1, a\}$) valuations, we prove that the Max-NSW allocation admits $\max\{1/a^2, 1/2\}$-EF1 and PO. Under strongly $p$-extendible system constraints, we show that the Max-NSW allocation guarantees $\max\{1/p, 1/4\}$-EF1 and PO for identical binary valuations. Indeed, the approximation of $1/4$ is the ratio for independence system constraints and additive valuations. Additionally, for lexicographic preferences, we study possibly feasible allocations other than Max-NSW admitting exactly EF1 and PO under the above constraints.

翻译：本研究探讨在拟阵约束及其两种推广形式——$p$-可扩展系统与独立系统约束下，具有加性估值智能体对不可分物品集合的公平分配问题。目标是寻找满足给定约束（即分配给每个智能体的物品子集符合约束条件）的公平高效分配方案。我们聚焦于经典的"除一物品无嫉妒"（EF1）公平性概念与高效（且公平）的"最大纳什社会福利"（Max-NSW）准则。通过运用拟阵性质，我们证明在拟阵约束下，满足帕累托最优（PO）的Max-NSW分配可实现紧致的$1/2$-EF1。该结果解决了既有文献[26]中提出的开放性问题。特别地，当智能体具有二值（$\{1, a\}$）估值时，我们证明Max-NSW分配可实现$\max\{1/a^2, 1/2\}$-EF1与PO。在强$p$-可扩展系统约束下，我们证明对于相同的二值估值，Max-NSW分配可保证$\max\{1/p, 1/4\}$-EF1与PO。实际上，$1/4$的近似比适用于独立系统约束与加性估值的情形。此外，针对字典序偏好，我们研究了在上述约束下除Max-NSW外可能存在的、能同时实现精确EF1与PO的可行分配方案。