We study the problem of fairly and efficiently allocating indivisible goods among agents with additive valuations. We focus on envy-freeness up to any good (EFX) -- an important fairness notion in fair division of indivisible goods. A central open question in this field is whether EFX allocations always exist for any number of agents. While prior work has established EFX existence for settings with at most three distinct valuations (Prakash HV et al. 2025) and for two types of goods (Gorantla, Marwaha, and Velusamy 2023), the general case remains unresolved. In this paper, we extend the existent knowledge by proving that EFX allocations satisfying Pareto optimality (PO) always exist and can be computed in quasiliniear time when there are two types of goods, given that the valuations are positive. This result strengthens the existing work of (Gorantla, Marwaha, and Velusamy 2023), which only guarantees the existence of EFX allocations without ensuring Pareto optimality. Our findings demonstrate a fairly simple and efficient algorithm constructing an EFX+PO allocation.

翻译：我们研究在具有可加性估值的主体间公平高效分配不可分割物品的问题。本文聚焦于"对任意物品无嫉妒"(EFX)这一不可分割物品公平分配领域的重要公平性概念。该领域的一个核心开放问题是：对于任意数量的主体，EFX分配是否始终存在。尽管已有研究证明了在至多三种不同估值情形下(Prakash HV等人，2025年)以及两类物品情形下(Gorantla、Marwaha和Velusamy，2023年)EFX分配的存在性，但一般情形仍未解决。本文通过证明当存在两类物品且估值均为正时，满足帕累托最优(PO)的EFX分配始终存在，并可在拟线性时间内计算，从而扩展了现有认知。该结果强化了(Gorantla、Marwaha和Velusamy，2023年)的现有工作——其仅保证EFX分配的存在性而未确保帕累托最优性。我们的研究展示了一种构造EFX+PO分配的相当简洁高效的算法。