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.
翻译:Schelling的模型考虑的是美元类型的代理商,其中每个代理商都需要在非方向图表上选择一个顶点,每个代理商都喜欢同一类型的近邻代理商。 我们的动机是最近的一项工作, 研究与代理商福利有关的理念的最佳解决方案。 我们探索计算这些解决方案的参数化复杂性。 我们注重的是经过仔细研究的社会福利和Pareto最佳性概念,以及最近提出的集团-福利最优性(GWO)和公用-维特最佳性(UVO)的概念,两者都位于WO和PO之间。 首先,我们关注的是基本案例,其中, 美元=2美元, 红色代理商和美元蓝色代理商。 我们展示了所有我们所考虑的解决方案-通知都是美元=1美元, 而当它们以美元和美元为基值的基值计算结果时, 我们用美元计算为基价的基价- 美元。