共同偏好图下不可分割物品分配中最小化最大不满度的研究
Minimizing Maximum Dissatisfaction in the Allocation of Indivisible Items under a Common Preference Graph
AI总结：
针对共同偏好图下的不可分割物品公平分配问题，研究最小化最大不满度的计算复杂度，给出多智能体场景的二分性结论及多种图结构下的多项式/固定参数可解结果。
摘要：
我们研究不可分割物品向智能体的分配任务，其中智能体对物品的偏好完全相同。偏好通过有向无环图刻画：顶点代表物品，边$(a,b)$表示每个智能体都更偏好物品$a$而非物品$b$。智能体的不满度由其未获得、且也未获得任何更受偏好物品的物品数量来衡量。研究目标是以公平方式分配物品，即最小化所有智能体中的最大不满度。我们研究了该问题的计算复杂度状态，并确立了如下二分性结论：当智能体数量至少为3时，即便在限制较强的图上，该问题也是NP难的；但针对双智能体场景可在多项式时间内求解。我们还针对不同的底层图结构给出了多项多项式时间可解的结果，例如宽度不超过2的图，以及星型图、匹配图等树状结构。此外，我们还补充了与路径模块、独立集模块相关的固定参数可解性结果。本文采用的技术包括瓶颈分配问题、贪心算法、动态规划、最大网络流以及整数线性规划。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
