Abstract
We consider the clustering with diversity problem: given a set of colored points in a metric space, partition them into clusters such that each cluster has at least ℓ points, all of which have distinct colors. We give a 2-approximation to this problem for any ℓ when the objective is to minimize the maximum radius of any cluster. We show that the approximation ratio is optimal unless P = NP, by providing a matching lower bound. Several extensions to our algorithm have also been developed for handling outliers. This problem is mainly motivated by applications in privacy-preserving data publication.
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References
Aggarwal, G., Feder, T., Kenthapadi, K., Khuller, S., Panigrahy, R., Thomas, D., Zhu, A.: Achieving anonymity via clustering. In: PODS, pp. 153–162 (2006)
Aggarwal, G., Feder, T., Kenthapadi, K., Motwani, R., Panigrahy, R., Thomas, D., Zhu, A.: Anonymizing tables. In: Eiter, T., Libkin, L. (eds.) ICDT 2005. LNCS, vol. 3363, pp. 246–258. Springer, Heidelberg (2004)
Ailon, N., Charikar, M., Newman, A.: Aggregating inconsistent information: Ranking and clustering. J. ACM 55(5), 1–27 (2008)
Arasu, A., Ré, C., Suciu, D.: Large-scale deduplication with constraints using Dedupalog. In: ICDE, pp. 952–963 (2009)
Bairoch, A., Apweiler, R.: The SWISS-PROT protein sequence data bank and its supplement TrEMBL. Nucleic acids research 25(1), 31 (1997)
Bansal, N., Blum, A., Chawla, S.: Correlation clustering. Machine Learning 56(1), 89–113 (2004)
Beresford, A., Stajano, F.: Location privacy in pervasive computing. IEEE Pervasive Computing, 46–55 (2003)
Wong, R.C.-W., Li, J., Fu, A.-C., Wang, K.: (α, k)-anonymity: an enhanced k-anonymity model for privacy preserving data publishing. In: SIGKDD, pp. 754–759 (2006)
Charikar, M., Khuller, S., Mount, D., Narasimhan, G.: Algorithms for facility location problems with outliers. In: SODA, pp. 642–651 (2001)
Davidson, I., Ravi, S.: Intractability and clustering with constraints. In: ICML, pp. 201–208 (2007)
Dwork, C., Naor, M., Reingold, O., Rothblum, G., Vadhan, S.: On the complexity of differentially private data release: efficient algorithms and hardness results. In: STOC, pp. 381–390 (2009)
Feldman, D., Fiat, A., Kaplan, H., Nissim, K.: Private coresets. In: STOC, pp. 361–370 (2009)
Ghinita, G., Karras, P., Kalnis, P., Mamoulis, N.: Fast data anonymization with low information loss. In: VLDB, pp. 758–769 (2007)
Giotis, I., Guruswami, V.: Correlation clustering with a fixed number of clusters. In: SODA, pp. 1176–1185 (2006)
Hoppner, F., Klawonn, F., Platz, R., Str, S.: Clustering with Size Constraints. Computational Intelligence Paradigms: Innovative Applications (2008)
Ji, X.: Graph Partition Problems with Minimum Size Constraints. PhD thesis, Rensselaer Polytechnic Institute (2004)
Kifer, D., Gehrke, J.: Injecting utility into anonymized datasets. In: SIGMOD, pp. 217–228 (2006)
Korte, B., Vygen, J.: Combinatorial Optimization: Theory and Algorithms, 4th edn. Springer, Heidelberg (2007)
LeFevre, K., DeWitt, D.J., Ramakrishnan, R.: Mondrian multidimensional k-anonymity. In: ICDE, p. 25 (2006)
Li, J., Yi, K., Zhang, Q.: Clustering with diversity (2010), http://arxiv.org/abs/1004.2968
Machanavajjhala, A., Gehrke, J., Kifer, D., Venkitasubramaniam, M.: l-diversity: Privacy beyond k-anonymity. In: ICDE, p. 24 (2006)
Meyerson, A., Williams, R.: On the complexity of optimal k-anonymity. In: PODS, pp. 223–228 (2004)
Alsuwaiyel, M.H.: Algorithms: Design Techniques and Analysis. World Scientific, Singapore (1998)
Park, H., Shim, K.: Approximate algorithms for k-anonymity. In: SIGMOD (2007)
Samarati, P.: Protecting respondents’ identities in microdata release. TKDE 13(6), 1010–1027 (2001)
Wagstaff, K., Cardie, C.: Clustering with instance-level constraints. In: ICML, pp. 1103–1110 (2000)
Wagstaff, K., Cardie, C., Schroedl, S.: Constrained k-means clustering with background knowledge. In: ICML, pp. 577–584 (2001)
Xiao, X., Tao, Y.: Anatomy: Simple and effective privacy preservation. In: VLDB, pp. 139–150 (2006)
Xiao, X., Tao, Y.: m-invariance: Towards privacy preserving re-publication of dynamic datasets. In: SIGMOD, pp. 689–700 (2007)
Xiao, X., Yi, K., Tao, Y.: The hardness and approximation algorithms for l-diversity. In: EDBT (2010)
Xing, E., Ng, A., Jordan, M., Russell, S.: Distance metric learning, with application to clustering with side-information. In: NIPS, pp. 505–512 (2003)
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Li, J., Yi, K., Zhang, Q. (2010). Clustering with Diversity. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds) Automata, Languages and Programming. ICALP 2010. Lecture Notes in Computer Science, vol 6198. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14165-2_17
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DOI: https://doi.org/10.1007/978-3-642-14165-2_17
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