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

TL;DR

This paper introduces FPT algorithms for min-max $r$-gather clustering and gathering problems on spider graphs, and proves their NP-hardness when the number of legs increases.

Contribution

It presents the first FPT algorithms parameterized by the number of legs and establishes NP-hardness for larger instances.

Findings

FPT algorithms are effective when the number of legs is small.

Problems are NP-hard for arbitrary number of legs.

First algorithms and hardness results for these problems on spider graphs.

Abstract

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