Approximation Algorithms for Clustering with Minimum Sum of Radii, Diameters, and Squared Radii

TL;DR

This paper introduces improved approximation algorithms for three clustering problems in metric spaces, achieving tighter bounds on the sum of radii, diameters, and squared radii, advancing the state of the art in clustering approximations.

Contribution

The paper presents new approximation algorithms with better guarantees for MSR, MSD, and MSSR clustering problems, improving previous bounds and providing tighter approximation ratios.

Findings

Achieved a 3.389-approximation for MSR

Achieved a 6.546-approximation for MSD

Developed an 11.078-approximation for MSSR

Abstract

In this paper, we present an improved approximation algorithm for three related problems. In the Minimum Sum of Radii clustering problem (MSR), we aim to select $k$ balls in a metric space to cover all points while minimizing the sum of the radii. In the Minimum Sum of Diameters clustering problem (MSD), we are to pick $k$ clusters to cover all the points such that sum of diameters of all the clusters is minimized. At last, in the Minimum Sum of Squared Radii problem (MSSR), the goal is to choose $k$ balls, similar to MSR. However in MSSR, the goal is to minimize the sum of squares of radii of the balls. We present a 3.389-approximation for MSR and a 6.546-approximation for MSD, improving over respective 3.504 and 7.008 developed by Charikar and Panigrahy (2001). In particular, our guarantee for MSD is better than twice our guarantee for MSR. In the case of MSSR, the best known approximation guarantee is $4\cdot(540)^{2}$ based on the work of Bhowmick, Inamdar, and Varadarajan in their general analysis of the $t$-Metric Multicover Problem. At last with our analysis, we get a 11.078-approximation algorithm for Minimum Sum of Squared Radii.