We study the fair division of indivisible items. In the general model, the goal is to allocate $m$ indivisible items to $n$ agents while satisfying fairness criteria such as MMS, EF1, and EFX. We also study a recently-introduced graphical model that represents the fair division problem as a multigraph, in which vertices correspond to agents and edges to items. The graphical model stipulates that an item can have non-zero marginal utility to an agent only if its corresponding edge is incident to the agent's corresponding vertex. We study orientations (allocations that allocate each edge to an endpoint) in this model, as they are particularly desirable. Our first contribution concerns MMS allocations of mixed manna (i.e. a mixture of goods and chores) in the general model. It is known that MMS allocations of goods exist when $m \leq n+5$. We generalize this and show that when $m \leq n+5$, MMS allocations of mixed manna exist as long as $n \leq 3$, there is an agent whose MMS threshold is non-negative, or every item is a chore. Remarkably, our result leaves only the case where every agent has a negative MMS threshold unanswered. Our second contribution concerns EFX orientations of multigraphs of goods. We show that deciding whether EFX orientations exist for multigraphs is NP-complete, even for symmetric bi-valued multigraphs. Complementarily, we show symmetric bi-valued multigraphs that do not contain non-trivial odd multitrees have EFX orientations that can be found in polynomial time. Our third contribution concerns EF1 and EFX orientations of graphs and multigraphs of chores. We obtain polynomial-time algorithms for deciding whether such graphs have EF1 and EFX orientations, resolving a previous conjecture and showing a fundamental difference between goods and chores division. In addition, we show that the analogous problems for multigraphs are NP-hard.

翻译：我们研究不可分割物品的公平分配问题。在一般模型中，目标是将 $m$ 件不可分割物品分配给 $n$ 个智能体，同时满足如 MMS、EF1 和 EFX 等公平性准则。我们还研究了一个近期提出的图模型，该模型将公平分配问题表示为多重图，其中顶点对应智能体，边对应物品。该图模型规定，仅当物品对应的边与智能体对应的顶点相关联时，该物品对该智能体才具有非零边际效用。我们研究该模型中的定向分配（即将每条边分配给其某一端点的分配方案），因其具有特殊优势。我们的第一项贡献涉及一般模型中混合物品（即物品与杂务的混合）的 MMS 分配。已知当 $m \leq n+5$ 时，物品的 MMS 分配存在。我们推广了这一结论，证明当 $m \leq n+5$ 时，只要满足以下条件之一，混合物品的 MMS 分配就存在：$n \leq 3$，存在 MMS 阈值为非负的智能体，或所有物品均为杂务。值得注意的是，我们的结果仅遗留了所有智能体的 MMS 阈值均为负值这一情形尚未解决。我们的第二项贡献涉及物品多重图的 EFX 定向分配。我们证明，判定多重图是否存在 EFX 定向分配是 NP 完全问题，即使对于对称双值多重图也是如此。作为补充，我们证明不包含非平凡奇数多重树的对称双值多重图存在可在多项式时间内找到的 EFX 定向分配。我们的第三项贡献涉及杂务图和多重图的 EF1 与 EFX 定向分配。我们获得了判定此类图是否存在 EF1 和 EFX 定向分配的多项式时间算法，从而证实了先前的猜想，并揭示了物品分配与杂务分配之间的根本差异。此外，我们证明多重图的类似问题是 NP 困难的。