PTAS and Exact Algorithms for $r$-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.