License: 
Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SWAT.2022.28
URN: urn:nbn:de:0030-drops-161881
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16188/
Inamdar, Tanmay ;
Varadarajan, Kasturi
Non-Uniform k-Center and Greedy Clustering
Abstract
In the Non-Uniform k-Center (NUkC) problem, a generalization of the famous k-center clustering problem, we want to cover the given set of points in a metric space by finding a placement of balls with specified radii. In t-NUkC, we assume that the number of distinct radii is equal to t, and we are allowed to use k_i balls of radius r_i, for 1 ≤ i ≤ t. This problem was introduced by Chakrabarty et al. [ACM Trans. Alg. 16(4):46:1-46:19], who showed that a constant approximation for t-NUkC is not possible if t is unbounded, assuming ? ≠ NP. On the other hand, they gave a bicriteria approximation that violates the number of allowed balls as well as the given radii by a constant factor. They also conjectured that a constant approximation for t-NUkC should be possible if t is a fixed constant. Since then, there has been steady progress towards resolving this conjecture - currently, a constant approximation for 3-NUkC is known via the results of Chakrabarty and Negahbani [IPCO 2021], and Jia et al. [SOSA 2022]. We push the horizon by giving an O(1)-approximation for the Non-Uniform k-Center for 4 distinct types of radii. Our result is obtained via a novel combination of tools and techniques from the k-center literature, which also demonstrates that the different generalizations of k-center involving non-uniform radii, and multiple coverage constraints (i.e., colorful k-center), are closely interlinked with each other. We hope that our ideas will contribute towards a deeper understanding of the t-NUkC problem, eventually bringing us closer to the resolution of the CGK conjecture.
BibTeX - Entry
@InProceedings{inamdar_et_al:LIPIcs.SWAT.2022.28,
author = {Inamdar, Tanmay and Varadarajan, Kasturi},
title = {{Non-Uniform k-Center and Greedy Clustering}},
booktitle = {18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022)},
pages = {28:1--28:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-236-5},
ISSN = {1868-8969},
year = {2022},
volume = {227},
editor = {Czumaj, Artur and Xin, Qin},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16188},
URN = {urn:nbn:de:0030-drops-161881},
doi = {10.4230/LIPIcs.SWAT.2022.28},
annote = {Keywords: k-center, approximation algorithms, non-uniform k-center, clustering}
}
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Keywords: |
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k-center, approximation algorithms, non-uniform k-center, clustering |
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Collection: |
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18th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2022) |
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Issue Date: |
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2022 |
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Date of publication: |
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22.06.2022 |
