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On Approximability of 𝓁₂² Min-Sum Clustering

Authors Karthik C. S. , Euiwoong Lee , Yuval Rabani , Chris Schwiegelshohn , Samson Zhou

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  • Theory of computation β†’ Computational geometry
  • Theory of computation β†’ Facility location and clustering
Keywords
  • Clustering
  • hardness of approximation
  • polynomial-time approximation schemes
  • learning-augmented algorithms

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Abstract

The 𝓁₂² min-sum k-clustering problem is to partition an input set into clusters C_1,…,C_k to minimize βˆ‘_{i=1}^k βˆ‘_{p,q ∈ C_i} β€–p-qβ€–β‚‚Β². Although 𝓁₂² min-sum k-clustering is NP-hard, it is not known whether it is NP-hard to approximate 𝓁₂² min-sum k-clustering beyond a certain factor. 
In this paper, we give the first hardness-of-approximation result for the 𝓁₂² min-sum k-clustering problem. We show that it is NP-hard to approximate the objective to a factor better than 1.056 and moreover, assuming a balanced variant of the Johnson Coverage Hypothesis, it is NP-hard to approximate the objective to a factor better than 1.327. 
We then complement our hardness result by giving a fast PTAS for 𝓁₂² min-sum k-clustering. Specifically, our algorithm runs in time O(n^{1+o(1)}dβ‹… 2^{(k/Ξ΅)^O(1)}), which is the first nearly linear time algorithm for this problem. We also consider a learning-augmented setting, where the algorithm has access to an oracle that outputs a label i ∈ [k] for input point, thereby implicitly partitioning the input dataset into k clusters that induce an approximately optimal solution, up to some amount of adversarial error Ξ± ∈ [0,1/2). We give a polynomial-time algorithm that outputs a (1+Ξ³Ξ±)/(1-Ξ±)Β²-approximation to 𝓁₂² min-sum k-clustering, for a fixed constant Ξ³ > 0.

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Karthik C. S., Euiwoong Lee, Yuval Rabani, Chris Schwiegelshohn, and Samson Zhou. On Approximability of 𝓁₂² Min-Sum Clustering. In 41st International Symposium on Computational Geometry (SoCG 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 332, pp. 62:1-62:18, Schloss Dagstuhl – Leibniz-Zentrum fΓΌr Informatik (2025) https://doi.org/10.4230/LIPIcs.SoCG.2025.62

Author Details

Karthik C. S.
  • Rutgers University, Piscataway, NJ, USA
Euiwoong Lee
  • University of Michigan, Ann Arbor, MI, USA
Yuval Rabani
  • The Hebrew University of Jerusalem, Israel
Chris Schwiegelshohn
  • Aarhus University, Denmark
Samson Zhou
  • Texas A&M University, College Station, TX, USA

Funding

  • Karthik C. S.: Supported by NSF grants CCF-2313372 and CCF-2443697, a grant from the Simons Foundation, Grant Number 825876, Awardee Thu D. Nguyen, and partially funded by the Ministry of Education and Science of Bulgaria’s support for INSAIT, Sofia University "St. Kliment Ohridski" as part of the Bulgarian National Roadmap for Research Infrastructure.
  • Lee, Euiwoong: Supported in part by NSF grant CCF-2236669 and Google.
  • Rabani, Yuval: Supported in part by ISF grants 3565-21 and 389-22, and by BSF grant 2023607.
  • Schwiegelshohn, Chris: Supported in part by the Independent Research Fund Denmark (DFF) under a Sapere Aude Research Leader grant No 1051-00106B and Google.
  • Zhou, Samson: Supported in part by NSF CCF-2335411. The work was conducted in part while the author was visiting the Simons Institute for the Theory of Computing as part of the Sublinear Algorithms program.

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