Abstract
In the k-means problem, we are given a finite set S of points in \(\Re^m\), and integer k ≥ 1, and we want to find k points (centers) so as to minimize the sum of the square of the Euclidean distance of each point in S to its nearest center. We show that this well-known problem is NP-hard even for instances in the plane, answering an open question posed by Dasgupta [6].
Part of the work by the third author was done when visiting The Institute of Mathematical Sciences, Chennai. He was also supported by NSF CAREER award CCR 0237431.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Aloise, D., Deshpande, A., Hansen, P., Popat, P.: NP-Hardness of Euclidean Sum-of-Squares Clustering. Technical Report G-2008-33, Les Cahiers du GERAD (to appear in Machine Learning) (April 2008)
Allender, E., Datta, S., Roy, S.: The directed planar reachability problem. In: Ramanujam, R., Sen, S. (eds.) FSTTCS 2005. LNCS, vol. 3821, pp. 238–249. Springer, Heidelberg (2005)
Arthur, D., Vassilvitskii, S.: Worst-case and smoothed analysis of the ICP algorithm, with an application to the k-means method. In: Proc. IEEE Symp. Foundations of Computer Science (2006)
Arthur, D., Vassilvitskii, S.: How slow is the k-means method? In: Proc. Symp. on Comput. Geom. (2006)
Arthur, D., Vassilvitskii, S.: k-means++: The advantages of careful seeding. In: Proc. ACM-SIAM Symp. Discrete Algorithms (2007)
Dasgupta, S.: The hardness of k-means clustering. Technical Report CS2007-0890, University of California, San Diego (2007)
de la Vega, F., Karpinski, M., Kenyon, C.: Approximation schemes for clustering problems. In: Proc. ACM Symp. Theory of Computing, pp. 50–58 (2003)
Drineas, P., Friexe, A., Kannan, R., Vempala, S., Vinay, V.: Clustering large graphs via the singular value decomposition. Machine Learning 56, 9–33 (2004)
Gibson, M., Kanade, G., Krohn, E., Pirwani, I., Varadarajan, K.: On clustering to minimize the sum of radii. In: Proc. ACM-SIAM Symp. Discrete Algorithms (2008)
Har-Peled, S., Sadri, B.: How fast is the k-means method? In: Proc. ACM-SIAM Symp. Discrete Algorithms, pp. 877–885 (2005)
Inaba, M., Katoh, N., Imai, H.: Applications of weighted Voronoi diagrams and randomization to variance-based clustering. In: Proc. Annual Symp. on Comput. Geom., pp. 332–339 (1994)
Kanungo, T., Mount, D., Netanyahu, N., Piatko, C., Silverman, R., Wu, A.: A local search approximation algorithm for k-means clustering. Comput. Geom. 28, 89–112 (2004)
Kumar, A., Sabharwal, Y., Sen, S.: A simple linear time (1 + ε) approximation algorithm for k-means clustering in any dimensions. In: Proc. IEEE Symp. Foundations of Computer Science, pp. 454–462 (2004)
Leiserson, C.E.: Area-efficient graph layouts (for VLSI). In: Proc. 21st Ann. IEEE Symp. Foundations of Computer Science, pp. 270–281 (1980)
Lichtenstein, D.: Planar formulae and their uses. SIAM J. Comput. 11, 329–343 (1982)
Lloyd, S.: Least squares quantization in PCM. IEEE Transactions on Information Theory 28, 129–136 (1982)
Megiddo, N., Supowit, K.: On the complexity of some common geometric location problems. SIAM J. Comput. 13, 182–196 (1984)
Ostrovsky, R., Rabani, Y., Schulman, L., Swamy, C.: The effectiveness of Lloyd-type methods for the k-means problem. In: Proc. IEEE Symp. Foundations of Computer Science (2006)
Valiant, L.G.: Universality considerations in VLSI circuits. IEEE Transactions on Computers 30, 135–140 (1981)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Mahajan, M., Nimbhorkar, P., Varadarajan, K. (2009). The Planar k-Means Problem is NP-Hard. In: Das, S., Uehara, R. (eds) WALCOM: Algorithms and Computation. WALCOM 2009. Lecture Notes in Computer Science, vol 5431. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00202-1_24
Download citation
DOI: https://doi.org/10.1007/978-3-642-00202-1_24
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00201-4
Online ISBN: 978-3-642-00202-1
eBook Packages: Computer ScienceComputer Science (R0)