We study four NP-hard optimal seat arrangement problems [Bodlaender et al., 2020a], which each have as input a set of n agents, where each agent has cardinal preferences over other agents, and an n-vertex undirected graph (called seat graph). The task is to assign each agent to a distinct vertex in the seat graph such that either the sum of utilities or the minimum utility is maximized, or it is envy-free or exchange-stable. Aiming at identifying hard and easy cases, we extensively study the algorithmic complexity of the four problems by looking into natural graph classes for the seat graph (e.g., paths, cycles, stars, or matchings), problem-specific parameters (e.g., the number of non-isolated vertices in the seat graph or the maximum number of agents towards whom an agent has non-zero preferences), and preference structures (e.g., non-negative or symmetric preferences). For strict preferences and seat graphs with disjoint edges and isolated vertices, we correct an error by Bodlaender et al. [2020b] and show that finding an envy-free arrangement remains NP-hard in this case.

翻译：我们研究了四个NP困难的座位安排最优化问题 [Bodlaender 等人，2020a]，每个问题的输入包括：一个包含n个智能体的集合（每个智能体对其他智能体具有基数偏好）、以及一个n顶点无向图（称为座位图）。任务是将每个智能体分配到座位图中的一个顶点（无重复分配），使得总效用或最小效用最大化，或实现无嫉妒性或交换稳定性。为识别困难与简单情形，我们通过考察座位图自然图类（如路径、环、星形或匹配图）、问题特定参数（如座位图中非孤立顶点数量或智能体对非零偏好的最大个数）以及偏好结构（如非负或对称偏好），广泛研究了四个问题的算法复杂性。对于严格偏好及座位图由不相交边与孤立顶点构成的情形，我们修正了Bodlaender 等人 [2020b] 中的一个错误，并证明在此情形下找到无嫉妒性安排仍是NP困难的。