A refined approximation for Euclidean k-means

Fabrizio Grandoni*, Rafail Ostrovsky, Yuval Rabani, Leonard J. Schulman, Rakesh Venkat

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

7 Scopus citations

Abstract

In the Euclidean k-Means problem we are given a collection of n points D in an Euclidean space and a positive integer k. Our goal is to identify a collection of k points in the same space (centers) so as to minimize the sum of the squared Euclidean distances between each point in D and the closest center. This problem is known to be APX-hard and the current best approximation ratio is a primal-dual 6.357 approximation based on a standard LP for the problem [Ahmadian et al. FOCS'17, SICOMP'20]. In this note we show how a minor modification of Ahmadian et al.'s analysis leads to a slightly improved 6.12903 approximation. As a related result, we also show that the mentioned LP has integrality gap at least [Formula presented].

Original languageAmerican English
Article number106251
Pages (from-to)1-7
Number of pages7
JournalInformation Processing Letters
Volume176
DOIs
StatePublished - Jun 2022

Bibliographical note

Publisher Copyright:
© 2022 The Author(s)

Keywords

  • Approximation algorithms
  • Euclidean facility location
  • Euclidean k-means
  • Integrality gaps

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