Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

37
The categories of this paper are: supr-con
15
The categories of this paper are: nucl-ex
26
The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
14
The categories of this paper are: math-ph
5
The categories of this paper are: gr-qc
28
The categories of this paper are: alg-geom
21
The categories of this paper are: adap-org
36
The categories of this paper are: cmp-lg
29
The categories of this paper are: funct-an
8
The categories of this paper are: hep-ex
16
The categories of this paper are: stat
17
The categories of this paper are: q-fin
9
The categories of this paper are: nlin
24
The categories of this paper are: dg-ga
34
The categories of this paper are: chem-ph