## 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 language | American English |
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Article number | 106251 |

Pages (from-to) | 1-7 |

Number of pages | 7 |

Journal | Information Processing Letters |

Volume | 176 |

DOIs | |

State | Published - Jun 2022 |

### Bibliographical note

Publisher Copyright:© 2022 The Author(s)

## Keywords

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