Article

Approximation schemes for clustering problems

Authors:

W. Fernandez de la Vega,

Marek Karpinski,

Yuval RabaniAuthors Info & Claims

STOC '03: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing

Pages 50 - 58

https://doi.org/10.1145/780542.780550

Published: 09 June 2003 Publication History

Abstract

Let k be a fixed integer. We consider the problem of partitioning an input set of points endowed with a distance function into k clusters. We give polynomial time approximation schemes for the following three clustering problems: Metric k-Clustering, l ²₂ k-Clustering, and l²₂ k-Median. In the k-Clustering problem, the objective is to minimize the sum of all intra-cluster distances. In the k-Median problem, the goal is to minimize the sum of distances from points in a cluster to the (best choice of) cluster center. In metric instances, the input distance function is a metric. In l ²₂ instances, the points are in R ^d and the distance between two points x,y is measured by x−y ²₂ (notice that (R ^d, ⋅ ²₂ is not a metric space). For the first two problems, our results are the first polynomial time approximation schemes. For the third problem, the running time of our algorithms is a vast improvement over previous work.

References

[1]

P. K. Agarwal and C. M. Procopiuc. Exact and approximation algorithms for clustering. In Proc. of the 9th Ann. ACM-SIAM Symp. on Discrete Algorithms, January 1998, pages 658--667.

Digital Library

[2]

N. Alon, S. Dar, M. Parnas, and D. Ron. Testing of clustering. In Proc. of the 41th Ann. IEEE Symp. on Foundations of Computer Science (FOCS) 2000, 240--250.

Digital Library

[3]

N. Alon and B. Sudakov. On two segmentation problems. Journal of Algorithms, 33:173--184, 1999.

Digital Library

[4]

S. Arora, D. Karger, and M. Karpinski. Polynomial time approximation schemes for dense instances of NP-hard problems. J. Comp. System. Sci., 58:193--210, 1999.

Digital Library

[5]

S. Arora, P. Raghavan, and S. Rao. Approximation schemes for Euclidean k-medians and related problems. In Proc. of the 30th Ann. ACM Symp. on Theory of Computing, 1998.

Digital Library

[6]

M. Bádoiu, S. Har-Peled, and P. Indyk. Approximate clustering via Core-Sets. Proc. 34th ACM STOC (2002), pages 250--257.

Digital Library

[7]

Y. Bartal, M. Charikar, and D. Raz. Approximating min-sum k-clustering in metric spaces. In Proc. of the 33rd Ann. ACM Symp. on Theory of Computing, July 2001, pages 11--20.

Digital Library

[8]

A. Broder, S. Glassman, M. Manasse, and G. Zweig. Syntactic clustering of the Web. In Proc. of the 6th Int'l World Wide Web Conf. (WWW), 1997, pages 391--404.

Digital Library

[9]

B. Carl and I. Stephani. Entropy, Compactness and the Approximation of Operators. Cambridge University Press, 1990.

[10]

M. Charikar and S. Guha. Improved combinatorial algorithms for the facility location and k-median problems. In Proc. of the 40th Ann. IEEE Symp. on Foundations of Computer Science, 1999.

Digital Library

[11]

M. Charikar, S. Guha, D.B. Shmoys, and É. Tardos. A constant factor approximation algorithm for the k-median problem. In Proc. of the 31st Ann. ACM Symp. on Theory of Computing, 1999.

Digital Library

[12]

S. Deerwester, S. T. Dumais, T. K. Landauer, G. W. Furnas, and R. A. Harshman. Indexing by latent semantic analysis. Journal of the Society for Information Science, 41(6):391--407, 1990.

[13]

W. Fernandez de la Vega, M. Karpinski, and C. Kenyon. A polynomial time approximation scheme for metric MIN-BISECTION. ECCC TR02-041, 2002.

Digital Library

[14]

W. Fernandez de la Vega and C. Kenyon. A randomized approximation scheme for metric MAX CUT. In Proc. of the 39th Ann. IEEE Symp. on Foundations of Computer Science (FOCS), 1998, pages 468--471, also in JCSS 63 (2001). pages 531--541.

Digital Library

[15]

P. Drineas, A. Frieze, R. Kannan, S. Vempala, and V. Vinay. Clustering in large graphs and matrices. In Proc. of the 10th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), 1999, pages 291--299.

Digital Library

[16]

C. Faloutsos, R. Barber, M. Flickner, J. Hafner, W. Niblack, D. Petkovic, and W. Equitz. Efficient and effective querying by image content. Journal of Intelligent Information Systems, 3(3):231--262, 1994.

Digital Library

[17]

N. Garg, V. Vazirani, and M. Yannakakis. Approximate max-flow min-(multi)cut theorems and their applications. SIAM Journal on Computing, 25(2):235--251, 1996.

Digital Library

[18]

O. Goldreich, S. Goldwasser, and D. Ron. Property testing and its connection to learning and approximation. J. of the ACM, 45:653--750, 1998.

Digital Library

[19]

S. Guha and S. Khuller. Greedy strikes back: Improved facility location algorithms. In Proc. of the 9th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), January 1998, 649--657.

Digital Library

[20]

N. Guttmann-Beck and R. Hassin. Approximation algorithms for min-sum p-clustering. Disc. Applied Math., 89:125--142, 1998.

Digital Library

[21]

P. Indyk. A sublinear time approximation scheme for clustering in metric spaces. In Proc. of the 40th Ann. IEEE Symp. on Foundations of Computer Science (FOCS), 1999, 154--159.

Digital Library

[22]

K. Jain and V.V. Vazirani. Primal-dual approximation algorithms for metric facility location and k-median problems. In Proc. of the 40th Ann. IEEE Symp. on Foundations of Computer Science, 1999.

Digital Library

[23]

V. Kann, S. Khanna, J. Lagergren, and A. Panconesi. On the hardness of approximating Max k-Cut and its dual. In Proc. of the 4th Israeli Symp. on Theory of Computing and Systems (ISTCS), 1996. Also in Chicago Journal of Theoretical Computer Science, 1997.

[24]

J. Kleinberg, C. Papadimitriou, and P. Raghavan. Segmentation problems. In Proc. of the 30th Ann. ACM Symp. on Theory of Computing (STOC), 1998, pages 473--482.

Digital Library

[25]

N. Mishra, D. Oblinger, and L. Pitt. Sublinear time approximate clustering. In Proc. of the 12th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA), January 2001, pages 439--447.

Digital Library

[26]

R. Ostrovsky and Y. Rabani. Polynomial time approximation schemes for geometric clustering problems. J. of the ACM, 49(2):139--156, March 2002.

Digital Library

[27]

S. Sahni and T. Gonzalez. P-complete approximation problems. Journal of the ACM, 23(3):555--565, 1976.

Digital Library

[28]

L. J. Schulman. Clustering for edge-cost minimization. In Proc. of the 32nd Ann. ACM Symp. on Theory of Computing (STOC), 2000, pages 547--555.

Digital Library

[29]

R. Shamir and R. Sharan. Algorithmic approaches to clustering gene expression data. In T. Jiang, T. Smith, Y. Xu, M.Q. Zhang eds., Current Topics in Computational Biology, MIT Press, to appear.

[30]

M. J. Swain and D. H. Ballard. Color indexing. International Journal of Computer Vision, 7:11--32, 1991.

Digital Library

Cited By

Bandyapadhyay SFriggstad ZMousavi R(2024)Parameterized Approximation Algorithms and Lower Bounds for k-Center Clustering and VariantsAlgorithmica10.1007/s00453-024-01236-186:8(2557-2574)Online publication date: 13-May-2024
https://doi.org/10.1007/s00453-024-01236-1
Feng QFu B(2024)Speeding Up Constrained k-Means Through 2-MeansAlgorithmic Aspects in Information and Management10.1007/978-981-97-7801-0_5(52-63)Online publication date: 19-Sep-2024
https://doi.org/10.1007/978-981-97-7801-0_5
Juyal AHosseini RNovikov DGrinshpon MZelikovsky A(2023)Reconstruction of Viral Variants via Monte Carlo ClusteringJournal of Computational Biology10.1089/cmb.2023.015430:9(1009-1018)Online publication date: 1-Sep-2023
https://doi.org/10.1089/cmb.2023.0154
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Index Terms

Approximation schemes for clustering problems
1. Theory of computation
  1. Randomness, geometry and discrete structures
    1. Computational geometry

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Approximation algorithms for clustering problems

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Published In

cover image ACM Conferences

STOC '03: Proceedings of the thirty-fifth annual ACM symposium on Theory of computing

June 2003

740 pages

ISBN:1581136749

DOI:10.1145/780542

Conference Chair:
Lawrence L. Larmore
University of Nevada Las Vegas, Las Vegas, NV
,
Program Chair:
Michel X. Goemans
Massachusetts Institute of Technology, Cambridge, MA

Copyright © 2003 ACM.

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Published: 09 June 2003

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STOC03: The 35th Annual ACM Symposium on Theory of Computing

June 9 - 11, 2003

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STOC '03 Paper Acceptance Rate 80 of 270 submissions, 30%;

Overall Acceptance Rate 1,469 of 4,586 submissions, 32%

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Cited By

Bandyapadhyay SFriggstad ZMousavi R(2024)Parameterized Approximation Algorithms and Lower Bounds for k-Center Clustering and VariantsAlgorithmica10.1007/s00453-024-01236-186:8(2557-2574)Online publication date: 13-May-2024
https://doi.org/10.1007/s00453-024-01236-1
Feng QFu B(2024)Speeding Up Constrained k-Means Through 2-MeansAlgorithmic Aspects in Information and Management10.1007/978-981-97-7801-0_5(52-63)Online publication date: 19-Sep-2024
https://doi.org/10.1007/978-981-97-7801-0_5
Juyal AHosseini RNovikov DGrinshpon MZelikovsky A(2023)Reconstruction of Viral Variants via Monte Carlo ClusteringJournal of Computational Biology10.1089/cmb.2023.015430:9(1009-1018)Online publication date: 1-Sep-2023
https://doi.org/10.1089/cmb.2023.0154
Juyal AHosseini RNovikov DGrinshpon MZelikovsky A(2022)Entropy Based Clustering of Viral SequencesBioinformatics Research and Applications10.1007/978-3-031-23198-8_33(369-380)Online publication date: 14-Nov-2022
https://dl.acm.org/doi/10.1007/978-3-031-23198-8_33
Cohen-Addad VKarthik CLee EMarx D(2021)On approximability of clustering problems without candidate centersProceedings of the Thirty-Second Annual ACM-SIAM Symposium on Discrete Algorithms10.5555/3458064.3458220(2635-2648)Online publication date: 10-Jan-2021
https://dl.acm.org/doi/10.5555/3458064.3458220
Fomin FGolovach PSimonov K(2021)Parameterized k-Clustering: Tractability islandJournal of Computer and System Sciences10.1016/j.jcss.2020.10.005117(50-74)Online publication date: May-2021
https://doi.org/10.1016/j.jcss.2020.10.005
Ghazi BKumar RManurangsi PLarochelle HRanzato MHadsell RBalcan MLin H(2020)Differentially private clusteringProceedings of the 34th International Conference on Neural Information Processing Systems10.5555/3495724.3496064(4040-4054)Online publication date: 6-Dec-2020
https://dl.acm.org/doi/10.5555/3495724.3496064
Kachouie NShutaywi M(2020)Weighted Mutual Information for Aggregated Kernel ClusteringEntropy10.3390/e2203035122:3(351)Online publication date: 18-Mar-2020
https://doi.org/10.3390/e22030351
Feldmann AKarthik C. S. KLee EManurangsi P(2020)A Survey on Approximation in Parameterized Complexity: Hardness and AlgorithmsAlgorithms10.3390/a1306014613:6(146)Online publication date: 19-Jun-2020
https://doi.org/10.3390/a13060146
Zhang YXia XXu XYu FWu HYu YWei B(2020)Robust Hierarchical Overlapping Community Detection With Personalized PageRankIEEE Access10.1109/ACCESS.2020.29988608(102867-102882)Online publication date: 2020
https://doi.org/10.1109/ACCESS.2020.2998860
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