享乐型座位安排问题
Hedonic Seat Arrangement Problems
AI总结：
本文提出享乐博弈变体 Seat Arrangement，分析其 PoF 界与四类安排的复杂性二分，并给出基于顶点覆盖数和交换次数的参数化复杂性结果。
摘要：
本文研究享乐博弈的一个变体，称为 \textsc{Seat Arrangement}（座位安排）。该模型通过一个双射来定义：把彼此具有偏好的智能体映射到图 $G$ 的顶点。一个智能体的效用取决于在图中分配给它的邻居。更确切地说，该效用是此智能体对分配到各邻居位置的智能体所持偏好的总和。我们首先考察不同偏好类别下的稳定价格与公平价格。特别地，我们证明存在一个实例，使公平价格（{\sf PoF}）在一般情况下无界。此外，我们给出 {\sf PoF} 的一个上界 $\tilde{d}(G)$ 以及一个几乎紧的下界 $\tilde{d}(G)-1/4$，其中 $\tilde{d}(G)$ 是输入图的平均度。随后，我们研究寻找某些“好”座位安排的问题的计算复杂性，即 \textsc{Utilitarian Arrangement}（功利主义安排）、\textsc{Egalitarian Arrangement}（平等主义安排）、\textsc{Stable Arrangement}（稳定安排）和 \textsc{Envy-free Arrangement}（无嫉妒安排）。我们从输入图连通分量的最大阶数这一角度，给出四个 \textsc{Seat Arrangement} 问题的计算复杂性二分。在参数化复杂性方面，\textsc{Utilitarian Arrangement} 可在 $n^{O(\gamma)}$ 时间内求解，但在 ETH 下不能在 $f(\gamma)n^{o(\gamma)}$ 时间内求解，其中 $n$ 是智能体数量，$\gamma$ 是输入图的顶点覆盖数。此外，我们证明即使在顶点覆盖数有界的图上，\textsc{Egalitarian Arrangement} 和 \textsc{Envy-free Arrangement} 也是弱 NP 难的。最后，我们证明：判断能否通过 $k$ 次交换从给定安排得到一个稳定安排，在以 $k+\gamma$ 为参数时是 W[1] 难的，但它可以在 $n^{O(k)}$ 时间内求解。
英文摘要：
In this paper, we study a variant of hedonic games, called \textsc{Seat Arrangement}. The model is defined by a bijection from agents with preferences for each other to vertices in a graph $G$. The utility of an agent depends on the neighbors assigned in the graph. More precisely, it is the sum over all neighbors of the preferences that the agent has towards the agent assigned to the neighbor. We first consider the price of stability and fairness for different classes of preferences. In particular, we show that there is an instance such that the price of fairness ({\sf PoF}) is unbounded in general. Moreover, we show an upper bound $\tilde{d}(G)$ and an almost tight lower bound $\tilde{d}(G)-1/4$ of {\sf PoF}, where $\tilde{d}(G)$ is the average degree of an input graph. Then we investigate the computational complexity of problems to find certain ``good'' seat arrangements, say \textsc{Utilitarian Arrangement}, \textsc{Egalitarian Arrangement}, \textsc{Stable Arrangement}, and \textsc{Envy-free Arrangement}. We give dichotomies of computational complexity of four \textsc{Seat Arrangement} problems from the perspective of the maximum order of connected components in an input graph. For the parameterized complexity, \textsc{Utilitarian Arrangement} can be solved in time $n^{O(γ)}$, while it cannot be solved in time $f(γ)n^{o(γ)}$ under ETH, where $n$ is the number of agents and $γ$ is the vertex cover number of an input graph. Moreover, we show that \textsc{Egalitarian Arrangement} and \textsc{Envy-free Arrangement} are weakly NP-hard even on graphs of bounded vertex cover number. Finally, we prove that determining whether a stable arrangement can be obtained from a given arrangement by $k$ swaps is W[1]-hard when parameterized by $k+γ$, whereas it can be solved in time $n^{O(k)}$.
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。
