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 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：我们考虑向智能体分配不可分物品的任务，其中智能体对物品的偏好相同。偏好通过有向无环图刻画，图中顶点表示物品，边$(a,b)$表示所有智能体都偏好物品$a$胜过物品$b$。智能体的不满程度由该智能体未获得且未获得任何更偏好物品的数量度量。目标是以公平方式向智能体分配物品，即最小化智能体间最大不满程度。我们研究了该问题的计算复杂性，并建立以下二分结果：当存在至少三个智能体时，即使在相当受限的图上问题也是NP难的，但两个智能体时可在多项式时间内求解。我们还针对不同底层图结构（如宽度不超过二的图、星形与匹配等树状结构）给出了若干多项式时间结果。这些发现与以路径模块和独立集模块为核心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。