We study the fair allocation of indivisible goods among agents, with a focus on limiting envy. A central open question in this area is the existence of EFX allocations-allocations in which any envy of any agent i towards any agent j vanishes upon the removal of any single good from j's bundle. Establishing the existence of such allocations has proven notoriously difficult in general, but progress has been made for restricted valuation classes. Christodoulou et al. [2023] proved existence for graphical valuations, where goods correspond to edges in a graph, agents to nodes, and each agent values only incident edges. The graph was required to be simple, i.e., for any pair of agents, there could be at most one good that both agents value. The problem remained open, however, for multi-graph valuations, where for a pair of agents several goods may have value to both. In this setting, Sgouritsa and Sotiriou [2025] established existence whenever the shortest cycle with non-parallel edges has length at least six, while Afshinmehr et al. [2025] proved existence when the graph contains no odd cycles. In this paper, we strengthen these results by proving that EFX allocations always exist in multi-graphs that contain no cycle of length three. Assuming monotone valuations, we further provide a pseudo-polynomial time algorithm for computing such an allocation, which runs in polynomial time when agents have cancelable valuations, a strict superclass of additive valuation functions. Accordingly, our results stand as one of the only cases where EFX allocations exist for an arbitrary number of agents.

翻译：本文研究不可分割物品在多个智能体间的公平分配问题，重点关注嫉妒限制。该领域的一个核心开放问题是EFX分配的存在性——即要求任意智能体i对任意智能体j的嫉妒，在移除j所获物品束中任意单个物品后即消失。已证明在一般情况下建立此类分配的存在性异常困难，但在受限估值类别中已取得进展。Christodoulou等人[2023]证明了图结构估值下的存在性，其中物品对应图中的边，智能体对应节点，且每个智能体仅估值与其关联的边。该研究要求图为简单图，即任意两个智能体之间至多存在一个双方都估值的物品。然而，对于多重图估值——即两个智能体之间可能存在多个双方都估值的物品——该问题仍未解决。在此设定下，Sgouritsa与Sotiriou[2025]证明了当不含平行边的最短环长度至少为六时存在EFX分配，而Afshinmehr等人[2025]则证明了图中不含奇数环时的存在性。本文通过证明在任意不包含长度为三的环（即无三角形）的多重图中EFX分配始终存在，强化了上述结果。在假设估值函数单调的条件下，我们进一步提出了一种计算此类分配的伪多项式时间算法，当智能体具有可抵消估值（即加性估值函数的严格超类）时，该算法可在多项式时间内运行。因此，我们的研究结果成为少数能证明任意数量智能体存在EFX分配的典型案例之一。