r-Gathering Problems on Spiders:Hardness, FPT Algorithms, and PTASes

TL;DR

This paper investigates the computational complexity of min-max r-gathering and clustering problems on spider metric spaces, establishing NP-hardness, developing fixed-parameter tractable algorithms, and providing approximation schemes.

Contribution

It proves NP-hardness on spider spaces, introduces FPT algorithms parameterized by the center degree, and develops PTASes for these problems.

Findings

NP-hardness on spider metric spaces

FPT algorithms parameterized by degree d

Existence of PTASes but no FPTASes unless P=NP

Abstract

We consider the min-max $r$-gathering problem described as follows: We are given a set of users and facilities in a metric space. We open some of the facilities and assign each user to an opened facility such that each facility has at least $r$ users. The goal is to minimize the maximum distance between the users and the assigned facility. We also consider the min-max $r$-gather clustering problem, which is a special case of the $r$-gathering problem in which the facilities are located everywhere. In this paper, we study the tractability and the hardness when the underlying metric space is a spider, which answers the open question posed by Ahmed et al. [WALCOM'19]. First, we show that the problems are NP-hard even if the underlying space is a spider. Then, we propose FPT algorithms parameterized by the degree $d$ of the center. This improves the previous algorithms because they are parameterized by both $r$ and $d$. Finally, we propose PTASes to the problems. These are best possible because there are no FPTASes unless P=NP.