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We consider min-max $r$-gather clustering problem and min-max $r$-gatheringproblem. In the min-max $r$-gather clustering problem, we are given a set ofusers and divide them into clusters with size at least $r$; the goal is tominimize the maximum diameter of clusters. In the min-max $r$-gatheringproblem, we are additionally given a set of facilities and assign each clusterto a facility; the goal is to minimize the maximum distance between the usersand the assigned facility. In this study, we consider the case that the usersand facilities are located on a ``spider'' and propose the firstfixed-parameter tractable (FPT) algorithms for both problems, which areparametrized by only the number of legs. Furthermore, we prove that theseproblems are NP-hard when the number of legs is arbitrarily large.

$r$-Gather Clustering and $r$-Gathering on Spider: FPT Algorithms and Hardness