Robust $k$-Center with Two Types of Radii

TL;DR

This paper presents a constant factor approximation algorithm for the non-uniform $k$-center problem with two radii types in the presence of outliers, overcoming previous integrality gap barriers using a novel ellipsoid method approach.

Contribution

It introduces a new ellipsoid-based technique to handle outliers in the two-radius $k$-center problem, advancing approximation algorithms in clustering.

Findings

Achieves constant factor approximation with outliers

Introduces a novel ellipsoid method within the ellipsoid method

Overcomes previous integrality gap barriers

Abstract

In the non-uniform $k$-center problem, the objective is to cover points in a metric space with specified number of balls of different radii. Chakrabarty, Goyal, and Krishnaswamy [ICALP 2016, Trans. on Algs. 2020] (CGK, henceforth) give a constant factor approximation when there are two types of radii. In this paper, we give a constant factor approximation for the two radii case in the presence of outliers. To achieve this, we need to bypass the technical barrier of bad integrality gaps in the CGK approach. We do so using "the ellipsoid method inside the ellipsoid method": use an outer layer of the ellipsoid method to reduce to stylized instances and use an inner layer of the ellipsoid method to solve these specialized instances. This idea is of independent interest and could be applicable to other problems. Keywords: Approximation, Clustering, Outliers, and Round-or-Cut.