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Near-Optimal Bounds for Parameterized Euclidean k-Means

Authors Vincent Cohen-Addad , Karthik C. S. , David Saulpic , Chris Schwiegelshohn

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ACM Subject Classification
  • Theory of computation → Facility location and clustering
  • Theory of computation → Computational geometry
Keywords
  • k-means clustering
  • Euclidean space
  • Fine-Grained Complexity

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Abstract

The k-means problem is a classic objective for modeling clustering in a metric space. Given a set of points in a metric space, the goal is to find k representative points so as to minimize the sum of the squared distances from each point to its closest representative. In this work, we study the approximability of k-means in Euclidean spaces parameterized by the number of clusters, k.
In seminal works, de la Vega, Karpinski, Kenyon, and Rabani [STOC'03] and Kumar, Sabharwal, and Sen [JACM'10] showed how to obtain a (1+ε)-approximation for high-dimensional Euclidean k-means in time 2^{(k/ε)^O(1)} ⋅ dn^O(1).
In this work, we introduce a new fine-grained hypothesis called Exponential Time for Expanders Hypothesis (XXH) which roughly asserts that there are no non-trivial exponential time approximation algorithms for the vertex cover problem on near perfect vertex expanders. Assuming XXH, we close the above long line of work on approximating Euclidean k-means by showing that there is no 2^{(k/ε)^{1-o(1)}} ⋅ n^O(1) time algorithm achieving a (1+ε)-approximation for k-means in Euclidean space. This lower bound is tight as it matches the algorithm given by Feldman, Monemizadeh, and Sohler [SoCG'07] whose runtime is 2^O(k/ε) + O(ndk). 
Furthermore, assuming XXH, we show that the seminal O(n^{kd+1}) runtime exact algorithm of Inaba, Katoh, and Imai [SoCG'94] for k-means is optimal for small values of k.

Cite As Get BibTex

Vincent Cohen-Addad, Karthik C. S., David Saulpic, and Chris Schwiegelshohn. Near-Optimal Bounds for Parameterized Euclidean k-Means. In 42nd International Symposium on Computational Geometry (SoCG 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 367, pp. 33:1-33:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SoCG.2026.33

Author Details

Vincent Cohen-Addad
  • Google Research, New York, NY, USA
Karthik C. S.
  • Rutgers University, Piscataway, NJ, USA
David Saulpic
  • Université Paris Cité, CNRS, IRIF, F-75013, Paris, France
Chris Schwiegelshohn
  • Aarhus University, Denmark

Funding

  • C. S., Karthik: This work was supported by the National Science Foundation under 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.
  • Schwiegelshohn, Chris: This work was partially supported by the Independent Research Fund Denmark (DFF) under a Sapere Aude Research Leader grant No 1051-00106B and by a Google Research Award.

Acknowledgements

We thank Dor Minzer, Euiwoong Lee, and Pasin Manurangsi for several discussions that helped conceptualize the proof approach in this paper.

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