Towards Non-Uniform k-Center with Constant Types of Radii

TL;DR

This paper advances the understanding of the Non-Uniform k-Center problem with a constant number of radii, providing a reduction technique and algorithms that improve approximation guarantees and simplify analysis.

Contribution

It introduces a black box reduction from t radii to t-1 radii with outliers, leading to a constant-factor approximation for three radii, and proposes a simplified bottom-up algorithm approach.

Findings

A reduction from t radii to t-1 radii with outliers.

A constant-factor approximation algorithm for three radii.

A simplified bottom-up algorithm with easier analysis.

Abstract

In the Non-Uniform k-Center problem we need to cover a finite metric space using k balls of different radii that can be scaled uniformly. The goal is to minimize the scaling factor. If the number of different radii is unbounded, the problem does not admit a constant-factor approximation algorithm but it has been conjectured that such an algorithm exists if the number of radii is constant. Yet, this is known only for the case of two radii. Our first contribution is a simple black box reduction which shows that if one can handle the variant of t-1 radii with outliers, then one can also handle t radii. Together with an algorithm by Chakrabarty and Negahbani for two radii with outliers, this immediately implies a constant-factor approximation algorithm for three radii, thus making further progress on the conjecture. Furthermore, using algorithms for the k-center with outliers problem, that is the one radii with outliers case, we also get a simple algorithm for two radii. The algorithm by Chakrabarty and Negahbani uses a top-down approach, starting with the larger radius and then proceeding to the smaller one. Our reduction, on the other hand, looks only at the smallest radius and eliminates it, which suggests that a bottom-up approach is promising. In this spirit, we devise a modification of the Chakrabarty and Negahbani algorithm which runs in a bottom-up fashion, and in this way we recover their result with the advantage of having a simpler analysis.