A refined approximation for Euclidean k-means

被引：8
|
作者：
Grandoni, Fabrizio [1 ]
Ostrovsky, Rafail [2 ]
Rabani, Yuval [3 ]
Schulman, Leonard J. [4 ]
Venkat, Rakesh [5 ]
机构：
[1] IDSIA, USI SUPSI, Lugano, Switzerland
[2] Univ Calif Los Angeles, Los Angeles, CA 90024 USA
[3] Hebrew Univ Jerusalem, Jerusalem, Israel
[4] CALTECH, Pasadena, CA 91125 USA
[5] IIT Hyderabad, Kandi, India
来源：
INFORMATION PROCESSING LETTERS | 2022年 / 176卷
基金：
美国国家科学基金会;
关键词：
Approximation algorithms; Euclidean k-means; Euclidean facility location; Integrality gaps; LOCAL SEARCH YIELDS;
D O I：
10.1016/j.ipl.2022.106251
中图分类号：
TP [自动化技术、计算机技术];
学科分类号：
0812 ;
摘要：
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 16+root 5 15 > 1.2157. (c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
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页数：7
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