  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.
