We study the computational complexity of fairly allocating a set of indivisible items under externalities. In this recently-proposed setting, in addition to the utility the agent gets from their bundle, they also receive utility from items allocated to other agents. We focus on the extended definitions of envy-freeness up to one item (EF1) and of envy-freeness up to any item (EFX), and we provide the landscape of their complexity for several different scenarios. We prove that it is NP-complete to decide whether there exists an EFX allocation, even when there are only three agents, or even when there are only six different values for the items. We complement these negative results by showing that when both the number of agents and the number of different values for items are bounded by a parameter the problem becomes fixed-parameter tractable. Furthermore, we prove that two-valued and binary-valued instances are equivalent and that EFX and EF1 allocations coincide for this class of instances. Finally, motivated from real-life scenarios, we focus on a class of structured valuation functions, which we term agent/item-correlated. We prove their equivalence to the ``standard'' setting without externalities. Therefore, all previous results for EF1 and EFX apply immediately for these valuations.

翻译：我们研究了在外部性条件下公平分配一组不可分割物品的计算复杂性。在这一近期提出的框架中，智能体不仅从自身所得物品组合中获得效用，还从分配给其他智能体的物品中获得效用。我们聚焦于扩展定义的无嫉妒性至多一件物品（EF1）和无嫉妒性至多任意一件物品（EFX），并展示了在不同场景下这些概念的复杂度全景。我们证明，即使仅有三个智能体，或物品价值仅有六种不同取值，判定是否存在EFX分配的问题也是NP完全的。为了平衡这些负面结果，我们指出当智能体数量和物品价值种类数均受某个参数限制时，该问题成为固定参数可处理的。此外，我们证明双值与二值实例是等价的，且在这类实例中EFX与EF1分配重合。最后，受现实场景启发，我们聚焦于一类结构化估值函数，称为智能体/物品相关函数，并证明了其与无外部性的“标准”设定等价。因此，所有针对EF1和EFX的已有结论均可直接适用于此类估值函数。