On Some Variants of Euclidean K-Supplier

TL;DR

This paper introduces improved approximation algorithms for Euclidean variants of the $k$-Supplier problem, achieving a factor of $(1+\sqrt{3})$, and establishes hardness results for the Matroid Supplier problem in Euclidean metrics.

Contribution

It provides the first known approximation algorithms better than 3 for Euclidean $k$-Supplier variants and shows hardness results for Euclidean Matroid Supplier.

Findings

Achieved $(1+\sqrt{3})$-approximation for Priority $k$-Supplier.

Achieved $(1+\sqrt{3})$-approximation for $k$-Supplier with Outliers.

Proved hardness of approximation for Euclidean Matroid Supplier to be at least 3.

Abstract

The $k$-Supplier problem is an important location problem that has been actively studied in both general and Euclidean metrics. Many of its variants have also been studied, primarily on general metrics. We study two variants of $k$-Supplier, namely Priority $k$-Supplier and $k$-Supplier with Outliers, in Euclidean metrics. We obtain $(1+\sqrt{3})$-approximation algorithms for both variants, which are the first improvements over the previously-known factor-$3$ approximation (that is known to be best-possible for general metrics). We also study the Matroid Supplier problem on Euclidean metrics, and show that it cannot be approximated to a factor better than $3$ (assuming $P\ne NP$); so the Euclidean metric offers no improvement in this case.