We study the problem of allocating $m$ indivisible goods among $n$ agents, where each agent's valuation is fractionally subadditive (XOS). With respect to AnyPrice Share (APS) fairness, Kulkarni et al. (2024) showed that, when agents have binary marginal values, a $0.1222$-APS allocation can be found in polynomial time, and there exists an instance where no allocation is better than $0.5$-approximate APS. Very recently, Feige and Grinberg (2025) extended the problem to the asymmetric case, where agents may have different entitlements, and improved the approximation ratio to $1/6$ for general XOS valuations. In this work, we focus on the asymmetric setting with binary XOS valuations, and further improve the approximation ratio to $1/2$, which matches the known upper bound. We also present a polynomial-time algorithm to compute such an allocation. Beyond APS fairness, we also study the weighted maximin share (WMMS) fairness. Farhadi et al. (2019) showed that, a $1/n$-WMMS allocation always exists for agents with general additive valuations, and that this approximation ratio is tight. We extend this result to general XOS valuations, where a $1/n$-WMMS allocation still exists, and this approximation ratio cannot be improved even when marginal values are binary. This shows a sharp contrast to binary additive valuations, where an exact WMMS allocation exists and can be found in polynomial time.

翻译：我们研究将$m$件不可分割物品分配给$n$个智能体的问题，其中每个智能体的估值函数为分数次可加（XOS）函数。在AnyPrice Share（APS）公平性方面，Kulkarni等人（2024）证明当智能体具有二元边际估值时，可在多项式时间内找到$0.1222$-APS分配，且存在实例表明不存在优于$0.5$近似APS的分配。最近，Feige与Grinberg（2025）将该问题扩展至非对称情形（智能体可具有不同权益），并将一般XOS估值的近似比提升至$1/6$。本文聚焦于具有二元XOS估值的非对称场景，进一步将近似比提升至$1/2$，该结果与已知上界匹配。我们同时提出了计算该分配的多项式时间算法。除APS公平性外，我们还研究了加权最大最小份额（WMMS）公平性。Farhadi等人（2019）证明对于具有一般可加估值的智能体，始终存在$1/n$-WMMS分配，且该近似比为紧界。我们将此结果扩展至一般XOS估值场景，证明$1/n$-WMMS分配依然存在，且即使边际估值为二元时该近似比亦不可改进。这与二元可加估值情形形成鲜明对比——后者存在精确WMMS分配且可在多项式时间内求解。