谢林隔离中福利保障的参数化复杂性
The Parameterized Complexity of Welfare Guarantees in Schelling Segregation
摘要：
谢林的模型考虑了$k$类型的代理，每个代理都需要在无向图上选择一个顶点，其中每个代理都喜欢相同类型的邻居代理。我们的动机是最近的一系列工作，研究的解决方案，是最佳的概念有关的福利代理。我们探讨计算这样的解决方案的参数化的复杂性。我们专注于社会福利（WO）和帕累托最优（PO）的研究概念，以及最近提出的群体福利最优（GWO）和效用向量最优（UVO）的概念，两者都位于WO和PO之间。首先，我们集中在基本情况下，其中$k=2$和有$r$红色代理和$b$蓝色代理。我们表明，所有的解决方案的概念，我们认为是$\textsf{NP}$ -难以计算，即使当$b=1$和它们是$\textsf{W}[1]$ -硬参数时，由$r$和$b$。此外，我们还证明了WO和GWO在三次图上也是$\textsf{NP}$ -硬的.我们补充这些负面的结果由$\textsf{FPT}$算法参数化的$r, b$和图的最大程度。对于$k$类型的代理的一般情况下，我们证明，对于任何的概念，我们认为这个问题是$\textsf{W}[1]$ -硬参数化时，$k$的一个大家庭的图形，包括树。我们伴随着这些负面的结果与$\textsf{XP}$算法参数化的$k$和树宽的图形。
英文摘要：
Schelling's model considers $k$ types of agents each of whom needs to select a vertex on an undirected graph, where every agent prefers to neighbor agents of the same type. We are motivated by a recent line of work that studies solutions that are optimal with respect to notions related to the welfare of the agents. We explore the parameterized complexity of computing such solutions. We focus on the well-studied notions of social welfare (WO) and Pareto optimality (PO), alongside the recently proposed notions of group-welfare optimality (GWO) and utility-vector optimality (UVO), both of which lie between WO and PO. Firstly, we focus on the fundamental case where $k=2$ and there are $r$ red agents and $b$ blue agents. We show that all solution-notions we consider are $\textsf{NP}$-hard to compute even when $b=1$ and that they are $\textsf{W}[1]$-hard when parameterized by $r$ and $b$. In addition, we show that WO and GWO are $\textsf{NP}$-hard even on cubic graphs. We complement these negative results by an $\textsf{FPT}$ algorithm parameterized by $r, b$ and the maximum degree of the graph. For the general case with $k$ types of agents, we prove that for any of the notions we consider the problem is $\textsf{W}[1]$-hard when parameterized by $k$ for a large family of graphs that includes trees. We accompany these negative results with an $\textsf{XP}$ algorithm parameterized by $k$ and the treewidth of the graph.
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