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（一种加性偏好的子域）下，不可分割物品与家务混合分配中的公平性问题。一个用于不可分割物品分配的著名公平概念是直至任意物品的无嫉妒性（EFX）。然而，其存在性与计算性一直是一个显著未解难题。通过识别一类包含"糟糕家务"的实例，我们证明判断EFX分配是否存在是NP完全的。这一结果直接暗示了在加性偏好下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集，对于任意混合实例，该类偏好下既保证存在EFX且帕累托最优（PO）的分配，又可高效计算。聚焦于两个较弱的公平概念，我们研究了在包含糟糕家务的特殊实例中寻找EF1和PO分配，并证明对于任意具有词典序偏好的混合实例，MMS和PO分配可被高效计算。