License: 
Creative Commons Attribution 3.0 Unported license (CC BY 3.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SOCG.2015.754
URN: urn:nbn:de:0030-drops-51178
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2015/5117/
Awasthi, Pranjal ;
Charikar, Moses ;
Krishnaswamy, Ravishankar ;
Sinop, Ali Kemal
The Hardness of Approximation of Euclidean k-Means
Abstract
The Euclidean k-means problem is a classical problem that has been extensively studied in the theoretical computer science, machine learning and the computational geometry communities. In this problem, we are given a set of n points in Euclidean space R^d, and the goal is to choose k center points in R^d so that the sum of squared distances of each point to its nearest center is minimized. The best approximation algorithms for this problem include a polynomial time constant factor approximation for general k and a (1+c)-approximation which runs in time poly(n) exp(k/c). At the other extreme, the only known computational complexity result for this problem is NP-hardness [Aloise et al.'09]. The main difficulty in obtaining hardness results stems from the Euclidean nature of the problem, and the fact that any point in R^d can be a potential center. This gap in understanding left open the intriguing possibility that the problem might admit a PTAS for all k, d.
In this paper we provide the first hardness of approximation for the Euclidean k-means problem. Concretely, we show that there exists a constant c > 0 such that it is NP-hard to approximate the k-means objective to within a factor of (1+c). We show this via an efficient reduction from the vertex cover problem on triangle-free graphs: given a triangle-free graph, the goal is to choose the fewest number of vertices which are incident on all the edges. Additionally, we give a proof that the current best hardness results for vertex cover can be carried over to triangle-free graphs. To show this we transform G, a known hard vertex cover instance, by taking a graph product with a suitably chosen graph H, and showing that the size of the (normalized) maximum independent set is almost exactly preserved in the product graph using a spectral analysis, which might be of independent interest.
BibTeX - Entry
@InProceedings{awasthi_et_al:LIPIcs:2015:5117,
author = {Pranjal Awasthi and Moses Charikar and Ravishankar Krishnaswamy and Ali Kemal Sinop},
title = {{The Hardness of Approximation of Euclidean k-Means}},
booktitle = {31st International Symposium on Computational Geometry (SoCG 2015)},
pages = {754--767},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-939897-83-5},
ISSN = {1868-8969},
year = {2015},
volume = {34},
editor = {Lars Arge and J{\'a}nos Pach},
publisher = {Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
address = {Dagstuhl, Germany},
URL = {http://drops.dagstuhl.de/opus/volltexte/2015/5117},
URN = {urn:nbn:de:0030-drops-51178},
doi = {10.4230/LIPIcs.SOCG.2015.754},
annote = {Keywords: Euclidean k-means, Hardness of Approximation, Vertex Cover}
}
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Keywords: |
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Euclidean k-means, Hardness of Approximation, Vertex Cover |
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Collection: |
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31st International Symposium on Computational Geometry (SoCG 2015) |
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Issue Date: |
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2015 |
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Date of publication: |
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12.06.2015 |
