Allocating indivisible items among a set of agents is a frequently studied discrete optimization problem. In the setting considered in this work, the agents' preferences over the items are assumed to be identical. We consider a very recent measure for the overall quality of an allocation which does not rely on numerical valuations of the items. Instead, it captures the agents' opinion by a directed acyclic preference graph with vertices representing items. An arc $(a,b)$ in such a graph means that the agents prefer item $a$ over item $b$. For a given allocation of items the dissatisfaction of an agent is defined as the number of items which the agent does not receive and for which no more preferred item is given to the agent. Our goal is to find an efficient allocation of the items to the agents such that the total dissatisfaction over all agents is minimized. We explore the dichotomy between NP-hard and polynomially solvable instances, depending on properties of the underlying preference graph. While the problem is NP-hard already for three agents even on very restricted graph classes, it is polynomially solvable for two agents on general preference graphs. For an arbitrary number of agents, we derive polynomial-time algorithms for relevant restrictions of the underlying undirected graph. These are trees and, among the graphs of treewidth two, series-parallel graphs and cactus graphs.

翻译：分配不可分物品给一组智能体是一个经常研究的离散优化问题。在本文考虑的情景中，假设智能体对物品的偏好是相同的。我们采用一种最新的整体分配质量衡量指标，该指标不依赖物品的数值估值，而是通过一个有向无环偏好图来捕捉智能体的意见，其中顶点代表物品。图中的一条弧$(a,b)$表示智能体更偏好物品$a$而非物品$b$。对于给定的物品分配，智能体的不满定义为：该智能体未获得的物品中，没有比已获得物品更优的物品的数量。我们的目标是找到一种高效的物品分配给所有智能体，使所有智能体的总不满最小化。我们探讨了NP难实例与多项式可解实例之间的二分性，这取决于底层偏好图的属性。即使在非常受限的图类上，该问题对三个智能体已是NP难的；而对于两个智能体，在一般偏好图上则是多项式可解的。对于任意数量的智能体，我们针对底层无向图的相关限制推导出了多项式时间算法，这些限制包括树，以及在树宽为2的图中，串并联图和仙人掌图。