PTAS and Exact Algorithms for rr-Gathering Problems on Tree

r-gathering problem is a variant of facility location problems. In this problem, we are given a set of users and a set of facilities on same metric space. We open some of the facilities and assign each user to an open facility, so that at least r users are assigned to every open facility. We aim to minimize the maximum distance between user and assigned facility. In general, this problem is NP-hard and admit an approximation algorithm with factor 3. It is known that the problem does not admit any approximation algorithm within a factor less than 3. In our another paper, we proved that this problem is NP-hard even on spider, which is a special case of tree metric. In this paper, we concentrate on the problems on a tree. First, we give a PTAS for r-gathering problem on a tree. Furthermore, we give PTAS for some variants of the problems on a tree, and also give exact polynomial-time algorithms for another variants of r-gathering problem on a tree

r-Gathering Problems on Spiders:Hardness, FPT Algorithms, and PTASes

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.Comment: This is work is a merger of arXiv:1907.04088 and arXiv:1907.0408