We consider the task of allocating indivisible items to agents, when the agents' preferences over the items are identical. The preferences are captured by means of a directed acyclic graph, with vertices representing items and an edge $(a,b)$, meaning that each of the agents prefers item $a$ over item $b$. The dissatisfaction of an agent is measured by the number of items that the agent does not receive and for which it also does not receive any more preferred item. The aim is to allocate the items to the agents in a fair way, i.e., to minimize the maximum dissatisfaction among the agents. We study the status of computational complexity of that problem and establish the following dichotomy: the problem is NP-hard for the case of at least three agents, even on fairly restricted graphs, but polynomially solvable for two agents. We also provide several polynomial-time results with respect to different underlying graph structures, such as graphs of width at most two and tree-like structures such as stars and matchings. These findings are complemented with fixed parameter tractability results related to path modules and independent set modules. Techniques employed in the paper include bottleneck assignment problem, greedy algorithm, dynamic programming, maximum network flow, and integer linear programming.

翻译：我们考虑将不可分物品分配给智能体的任务，其中智能体对物品的偏好相同。偏好通过有向无环图表示，顶点代表物品，边$(a,b)$表示所有智能体都偏好物品$a$胜过物品$b$。智能体的不满度定义为：该智能体未获得且未获得任何更偏好物品的数量。目标是公平地将物品分配给智能体，即最小化智能体中的最大不满度。我们研究该问题的计算复杂性状态，并建立以下二分法：对于至少三个智能体的情况，即使在相当受限的图上，该问题也是NP难的，但对于两个智能体则存在多项式时间解法。我们还提供了关于不同底层图结构的若干多项式时间结果，例如宽度最多为二的图以及星形和匹配等树状结构。这些发现与关于路径模和独立集模的固定参数可解性结果互为补充。本文采用的技术包括瓶颈分配问题、贪心算法、动态规划、最大网络流和整数线性规划。