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

Zachary Friggstad, Mahya Jamshidian

TL;DR

The paper addresses clustering with three closely related objectives: MSR (min-sum of radii), MSD (min-sum of diameters), and MSSR (min-sum of squared radii) using at most $k$ centers. It introduces a bi-point LP-rounding framework based on a common Lagrangian multiplier, combined with a three-phase MSR procedure (guess largest balls, obtain a bi-point via LP rounding, and merge bi-point solutions) to achieve a $3.389$-approximation for MSR, which in turn yields a $6.546$-approximation for MSD and an $11.078$-approximation for MSSR. The approach improves on prior constants (e.g., $3.504$ for MSR and $7.008$ for MSD by Charikar & Panigrahy) and provides a modular method that relies on LP relaxations and bi-point solutions, with a simple LMP variant via direct LP rounding. Supporting results include a binary-search-based bi-point construction and a GROUP-merge LP to control the number of centers while bounding the objective, with the analysis leveraging disjoint-rounding properties and careful case-based radius bounds. The findings advance constant-factor guarantees for center-based clustering and offer a framework potentially extensible to related multicover and squared-radius settings.

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.

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

TL;DR

The paper addresses clustering with three closely related objectives: MSR (min-sum of radii), MSD (min-sum of diameters), and MSSR (min-sum of squared radii) using at most centers. It introduces a bi-point LP-rounding framework based on a common Lagrangian multiplier, combined with a three-phase MSR procedure (guess largest balls, obtain a bi-point via LP rounding, and merge bi-point solutions) to achieve a -approximation for MSR, which in turn yields a -approximation for MSD and an -approximation for MSSR. The approach improves on prior constants (e.g., for MSR and for MSD by Charikar & Panigrahy) and provides a modular method that relies on LP relaxations and bi-point solutions, with a simple LMP variant via direct LP rounding. Supporting results include a binary-search-based bi-point construction and a GROUP-merge LP to control the number of centers while bounding the objective, with the analysis leveraging disjoint-rounding properties and careful case-based radius bounds. The findings advance constant-factor guarantees for center-based clustering and offer a framework potentially extensible to related multicover and squared-radius settings.

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 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 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 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 based on the work of Bhowmick, Inamdar, and Varadarajan in their general analysis of the -Metric Multicover Problem. At last with our analysis, we get a 11.078-approximation algorithm for Minimum Sum of Squared Radii.

Paper Structure

This paper contains 13 sections, 15 theorems, 29 equations, 1 figure, 4 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Our Contribution
  3. Other Related Work
  4. Minimum Sum of Radii
  5. Step 1: Guessing the Largest Balls
  6. Step 2: Getting a Bi-point Solution
  7. Step 3: Combining Bi-point Solutions for MSR
  8. Minimum Sum of Diameters
  9. Minimum Sum of Squared Radii
  10. A Simple LMP Algorithm via Direct LP Rounding
  11. Binary Search Preparation: Starting Values and Filling a Solution
  12. The Binary Search
  13. Supporting Results for the Binary Search

Key Result

Theorem 1

There is a polynomial-time 3.389-approximation for MSR.

Figures (1)

  • Figure 1: A depiction of a group $G_{(i_1,R_1)}$. The solid ball is $B(i_1,R_1)$ and the dashed balls are those in $G_{(i_1,R_1)}$. Point $j$ is covered by tripling the ball centered at $i'$. The dashed path depicts the way we bound $d(j,i_2)$ in the second part of the case Centering at $i_2$.

Theorems & Definitions (28)

  • Definition 1
  • Definition 2
  • Definition 3
  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • Lemma 1
  • proof
  • Theorem 5
  • ...and 18 more