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r-gathering problem is a variant of facility location problems. In thisproblem, we are given a set of users and a set of facilities on same metricspace. 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 tominimize the maximum distance between user and assigned facility. In general,this problem is NP-hard and admit an approximation algorithm with factor 3. Itis known that the problem does not admit any approximation algorithm within afactor less than 3. In our another paper, we proved that this problem isNP-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-gatheringproblem on a tree. Furthermore, we give PTAS for some variants of the problemson a tree, and also give exact polynomial-time algorithms for another variantsof r-gathering problem on a tree.

PTAS and Exact Algorithms for $r$-Gathering Problems on Tree