We study the fundamental problem of fairly allocating a multiset $\mathcal{M}$ of $t$ types of indivisible items among $d$ groups of agents, where all agents within a group have identical additive valuations. Gorantla et al. [GMV23] showed that for every such instance, there exists a finite number $μ$ such that, if each item type appears at least $μ$ times, an envy-free allocation exists. Their proof is non-constructive and only provides explicit upper bounds on $μ$ for the cases of two groups ($d=2$) or two item types ($t=2$). In this work, we resolve one of the main open questions posed by Gorantla et al. [GMV23] by deriving explicit upper bounds on $μ$ that hold for arbitrary numbers of groups and item types. We introduce a significantly simpler, yet powerful technique that not only yields constructive guarantees for indivisible goods but also extends naturally to chores and continuous domains, leading to new results in related fair division settings such as cake cutting.

翻译：我们研究一个基础性问题：如何将包含$t$种不可分物品的多重集合$\mathcal{M}$公平分配给$d$组智能体，其中同组内所有智能体具有相同的可加估值。Gorantla等人[GMV23]证明，对于任意此类实例，存在一个有限数$μ$，使得当每种物品至少出现$μ$次时，必存在无嫉妒分配。其证明是非构造性的，且仅针对两组智能体（$d=2$）或两种物品类型（$t=2$）的情形给出了$μ$的显式上界。本文通过推导适用于任意组数与物品类型数量的显式$μ$上界，解决了Gorantla等人[GMV23]提出的一个核心开放性问题。我们引入了一种显著简化却强有力的技术，该技术不仅能为不可分物品提供构造性保证，还能自然地推广至杂务分配与连续域，进而在蛋糕切割等相关公平分配场景中引出一系列新结果。