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Hybrid k-Clustering: Blending k-Median and k-Center

Authors Fedor V. Fomin , Petr A. Golovach , Tanmay Inamdar , Saket Saurabh , Meirav Zehavi

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Author Details

Fedor V. Fomin
  • University of Bergen, Norway
Petr A. Golovach
  • University of Bergen, Norway
Tanmay Inamdar
  • Indian Institute of Technology Jodhpur, Jodhpur, India
Saket Saurabh
  • The Institute of Mathematical Sciences, HBNI, Chennai, India
  • University of Bergen, Norway
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel

Funding

  • Fomin, Fedor V.: Supported by the Research Council of Norway via the project BWCA (grant no. 314528).
  • Golovach, Petr A.: Supported by the Research Council of Norway via the project BWCA (grant no. 314528).
  • Saurabh, Saket: The author is supported by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 819416); and he also acknowledges the support of Swarnajayanti Fellowship grant DST/SJF/MSA-01/2017-18.
  • Zehavi, Meirav: The research was supported by the European Research Council (ERC) grant no. 101039913 (PARAPATH).

Cite AsGet BibTex

Fedor V. Fomin, Petr A. Golovach, Tanmay Inamdar, Saket Saurabh, and Meirav Zehavi. Hybrid k-Clustering: Blending k-Median and k-Center. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 317, pp. 4:1-4:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.APPROX/RANDOM.2024.4

Abstract

We propose a novel clustering model encompassing two well-known clustering models: k-center clustering and k-median clustering. In the Hybrid k-Clustering problem, given a set P of points in ℝ^d, an integer k, and a non-negative real r, our objective is to position k closed balls of radius r to minimize the sum of distances from points not covered by the balls to their closest balls. Equivalently, we seek an optimal L₁-fitting of a union of k balls of radius r to a set of points in the Euclidean space. When r = 0, this corresponds to k-median; when the minimum sum is zero, indicating complete coverage of all points, it is k-center. Our primary result is a bicriteria approximation algorithm that, for a given ε > 0, produces a hybrid k-clustering with balls of radius (1+ε)r. This algorithm achieves a cost at most 1+ε of the optimum, and it operates in time 2^{(kd/ε)^𝒪(1)} ⋅ n^𝒪(1). Notably, considering the established lower bounds on k-center and k-median, our bicriteria approximation stands as the best possible result for Hybrid k-Clustering.

Subject Classification

ACM Subject Classification
  • Theory of computation → Facility location and clustering
  • Theory of computation → Fixed parameter tractability
Keywords
  • clustering
  • k-center
  • k-median
  • Euclidean space
  • fpt approximation

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