  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.
