A refined approximation for Euclidean k-means

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

doi:10.1016/j.ipl.2022.106251

A refined approximation for Euclidean k-means

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

^*Corresponding author for this work

The Rachel and Selim Benin School of Engineering and Computer Science

Research output: Contribution to journal › Article › peer-review

13 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 language	English
Article number	106251
Pages (from-to)	1-7
Number of pages	7
Journal	Information Processing Letters
Volume	176
DOIs	https://doi.org/10.1016/j.ipl.2022.106251
State	Published - 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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10.1016/j.ipl.2022.106251

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title = "A refined approximation for Euclidean k-means",

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].",

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author = "Fabrizio Grandoni and Rafail Ostrovsky and Yuval Rabani and Schulman, {Leonard J.} and Rakesh Venkat",

note = "Publisher Copyright: {\textcopyright} 2022 The Author(s)",

year = "2022",

month = jun,

doi = "10.1016/j.ipl.2022.106251",

language = "אנגלית",

volume = "176",

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T1 - A refined approximation for Euclidean k-means

AU - Grandoni, Fabrizio

AU - Ostrovsky, Rafail

AU - Rabani, Yuval

AU - Schulman, Leonard J.

AU - Venkat, Rakesh

PY - 2022/6

Y1 - 2022/6

N2 - 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].

AB - 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].

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KW - Integrality gaps

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A refined approximation for Euclidean k-means

Abstract

Bibliographical note

Keywords

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