We study the fair allocation of mixtures of indivisible goods and chores under lexicographic preferences$\unicode{x2014}$a subdomain of additive preferences. A prominent fairness notion for allocating indivisible items is envy-freeness up to any item (EFX). Yet, its existence and computation has remained a notable open problem. By identifying a class of instances with "terrible chores", we show that determining the existence of an EFX allocation is NP-complete. This result immediately implies the intractability of EFX under additive preferences. Nonetheless, we propose a natural subclass of lexicographic preferences for which an EFX and Pareto optimal (PO) allocation is guaranteed to exist and can be computed efficiently for any mixed instance. Focusing on two weaker fairness notions, we investigate finding EF1 and PO allocations for special instances with terrible chores, and show that MMS and PO allocations can be computed efficiently for any mixed instance with lexicographic preferences.

翻译：我们研究在词典序偏好（可加偏好的子域）下不可分割物品与家务混合物的公平分配问题。分配不可分割物品的一个显著公平性概念是“任意物品无嫉妒性”（EFX）。然而，其存在性与计算性一直是一个悬而未决的重要问题。通过识别一类包含“恶劣家务”的实例，我们证明判定是否存在EFX分配是NP完全的。这一结果直接暗示了在可加偏好下EFX的难解性。尽管如此，我们提出了一类自然的词典序偏好子类，在该子类下，EFX与帕累托最优（PO）分配保证存在且可针对任何混合实例高效计算。聚焦于两种较弱的公平性概念，我们研究了在包含恶劣家务的特殊实例中寻找EF1与PO分配的问题，并证明对于任何词典序偏好的混合实例，MMS与PO分配均可高效计算。