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<p>We present an improved approximation algorithm for three related clustering problems. In the Minimum Sum of Radii clustering problem (MSR), we are to select <i>k</i> balls in a metric space to cover all points while minimizing the sum of their radii. In the Minimum Sum of Diameters clustering problem (MSD), we are to pick <i>k</i> clusters to cover all the points such that sum of diameters of all the clusters is minimized. Finally, in the Minimum Sum of Squared Radii problem (MSSR), the goal is to choose <i>k</i> balls, similar to MSR but 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. Note that since our result, Buchem et al. proved a <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(3+\epsilon \)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>3</mn> <mo>+</mo> <mi>ϵ</mi> </mrow> </math></EquationSource> </InlineEquation>-approximation for MSR. In the case of MSSR, the best known approximation guarantee is <InlineEquation ID="IEq2"> <EquationSource Format="TEX">\(4\cdot (540)^{2}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mn>4</mn> <mo>·</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mn>540</mn> <mo stretchy="false">)</mo> </mrow> <mn>2</mn> </msup> </mrow> </math></EquationSource> </InlineEquation> based on the work of Bhowmick, Inamdar, and Varadarajan in their general analysis of the <i>t</i>-Metric Multicover Problem. Furthermore, our analysis yields an 11.078-approximation algorithm for the Minimum Sum of Squared Radii problem</p>

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

  • 期刊论文

  • 发表日期：

    2026年3月31日

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  • Zachary Friggstad,
  • Mahya Jamshidian

摘要

We present an improved approximation algorithm for three related clustering problems. In the Minimum Sum of Radii clustering problem (MSR), we are to select k balls in a metric space to cover all points while minimizing the sum of their 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. Finally, in the Minimum Sum of Squared Radii problem (MSSR), the goal is to choose k balls, similar to MSR but 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. Note that since our result, Buchem et al. proved a \(3+\epsilon \) 3 + ϵ -approximation for MSR. In the case of MSSR, the best known approximation guarantee is \(4\cdot (540)^{2}\) 4 · ( 540 ) 2 based on the work of Bhowmick, Inamdar, and Varadarajan in their general analysis of the t-Metric Multicover Problem. Furthermore, our analysis yields an 11.078-approximation algorithm for the Minimum Sum of Squared Radii problem