Spider上的r-聚集问题：困难性、FPT算法和PTASes
r-Gathering Problems on Spiders:Hardness, FPT Algorithms, and PTASes
摘要：
我们考虑的最小-最大$r$收集问题描述如下：我们给出了一组用户和设施的度量空间。我们打开一些协作室，并将每个用户分配给已打开的协作室，以便每个协作室至少有$r$用户。目标是最小化用户与指定设施之间的最大距离。我们还考虑了最小-最大$r$收集聚类问题，这是一个特殊的情况下，$r$收集问题中的设施位于无处不在。在本文中，我们研究的易处理性和硬度时，基本的度量空间是一个蜘蛛，这回答了艾哈迈德等人提出的公开问题。首先，我们表明，即使底层空间是一个蜘蛛的问题是NP-难的。然后，我们提出了FPT算法参数化的程度$d$的中心。这改进了先前的算法，因为它们由$r$和$d$两者参数化。最后，我们提出了PTASes的问题。这是最好的可能性，因为除非P=NP，否则没有FPTAS。
英文摘要：
We consider the min-max $r$-gathering problem described as follows: We are given a set of users and facilities in a metric space. We open some of the facilities and assign each user to an opened facility such that each facility has at least $r$ users. The goal is to minimize the maximum distance between the users and the assigned facility. We also consider the min-max $r$-gather clustering problem, which is a special case of the $r$-gathering problem in which the facilities are located everywhere. In this paper, we study the tractability and the hardness when the underlying metric space is a spider, which answers the open question posed by Ahmed et al. [WALCOM'19]. First, we show that the problems are NP-hard even if the underlying space is a spider. Then, we propose FPT algorithms parameterized by the degree $d$ of the center. This improves the previous algorithms because they are parameterized by both $r$ and $d$. Finally, we propose PTASes to the problems. These are best possible because there are no FPTASes unless P=NP.
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。
