Abstract
Clustering large data is a fundamental task with widespread applications. The distributed computation methods have received greatly attention in recent years due to the increasing size of data. In this paper, we consider a variant of the widely used k-center problem, i.e., the lower-bounded k-center problem, and study the lower-bounded k-center problem in the Massively Parallel Computation (MPC) model. The lower-bounded k-center problem takes as input a set C of points in a metric space, the desired number k of centers, and a lower bound L. The goal is to partition the set C into at most k clusters such that the number of points in each cluster is at least L, and the k-center clustering objective is minimized. The current best result for the above problem in the MPC model is 16-approximation algorithm with 4 rounds. In this paper, we obtain a 2-round \((7+\epsilon )\)-approximation algorithm for this problem in the MPC model.
This work was supported by National Natural Science Foundation of China (62172446), Open Project of **angjiang Laboratory (22XJ02002), and Central South University Research Programme of Advanced Interdisciplinary Studies (2023QYJC023).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Aggarwal, G., et al.: Achieving anonymity via clustering. ACM Trans. Algorithms 6(3), 49:1–49:19 (2010)
Aghamolaei, S., Ghodsi, M., Miri, S.: A mapreduce algorithm for metric anonymity problems. In: Proceedings of the 31st Canadian Conference on Computational Geometry, pp. 117–123 (2019)
Ahmadian, S., Swamy, C.: Approximation algorithms for clustering problems with lower bounds and outliers. In: Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming, pp. 69:1–69:15 (2016)
An, H., Bhaskara, A., Chekuri, C., Gupta, S., Madan, V., Svensson, O.: Centrality of trees for capacitated \(k\)-center. Math. Program. 154(1–2), 29–53 (2015)
Angelidakis, H., Kurpisz, A., Sering, L., Zenklusen, R.: Fair and fast \(k\)-center clustering for data summarization. In: Proceedings of the 39th International Conference on Machine Learning, pp. 669–702 (2022)
Chakrabarty, D., Goyal, P., Krishnaswamy, R.: The non-uniform \(k\)-center problem. In: Proceedings of the 43rd International Colloquium on Automata, Languages, and Programming, pp. 67:1–67:15 (2016)
Charikar, M., Khuller, S., Mount, D.M., Narasimhan, G.: Algorithms for facility location problems with outliers. In: Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 642–651 (2001)
Cygan, M., Hajiaghayi, M., Khuller, S.: LP rounding for \(k\)-centers with non-uniform hard capacities. In: Proceedings of the 53rd Annual Symposium on Foundations of Computer Science, pp. 273–282 (2012)
Epasto, A., Mahdian, M., Mirrokni, V., Zhong, P.: Massively parallel and dynamic algorithms for minimum size clustering. In: Proceedings of the 33rd Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1613–1660 (2022)
Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. WH Freeman, New York (1979)
Gonzalez, T.F.: Clustering to minimize the maximum intercluster distance. Theoret. Comput. Sci. 38, 293–306 (1985)
Hansen, P., Brimberg, J., Urošević, D., Mladenović, N.: Solving large p-median clustering problems by primal-dual variable neighborhood search. Data Min. Knowl. Disc. 19, 351–375 (2009)
Haqi, A., Zarrabi-Zadeh, H.: Almost optimal massively parallel algorithms for k-center clustering and diversity maximization. In: Proceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures, pp. 239–247 (2023)
Hochbaum, D.S., Shmoys, D.B.: A best possible heuristic for the \(k\)-center problem. Math. Oper. Res. 10(2), 180–184 (1985)
Ip, W., Mou, W.: Customer grou** for better resources allocation using GA based clustering technique. Expert Syst. Appl. 39(2), 1979–1987 (2012)
Jones, M., Lê Nguyên, H., Nguyen, T.: Fair \(k\)-centers via maximum matching. In: Proceedings of the 37th International Conference on Machine Learning, pp. 4940–4949 (2020)
Kleindessner, M., Samadi, S., Awasthi, P., Morgenstern, J.: Guarantees for spectral clustering with fairness constraints. In: Proceedings of the 36th International Conference on Machine Learning, pp. 3458–3467 (2019)
Malkomes, G., Kusner, M.J., Chen, W., Weinberger, K.Q., Moseley, B.: Fast distributed \(k\)-center clustering with outliers on massive data. In: Advances in Neural Information Processing Systems, vol. 28, pp. 1063–1071 (2015)
Pan, W., Shen, X., Liu, B.: Cluster analysis: unsupervised learning via supervised learning with a non-convex penalty. J. Mach. Learn. Res. 14(7), 1865–1889 (2013)
Rollet, R., Benie, G., Li, W., Wang, S., Boucher, J.: Image classification algorithm based on the RBF neural network and \(k\)-means. Int. J. Remote Sens. 19(15), 3003–3009 (1998)
Rösner, C., Schmidt, M.: Privacy preserving clustering with constraints. In: Proceedings of the 45th International Colloquium on Automata, Languages, and Programming, pp. 96:1–96:14 (2018)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd.
About this paper
Cite this paper
Liang, T., Feng, Q., Wu, X., Xu, J., Wang, J. (2024). Improved Approximation Algorithm for the Distributed Lower-Bounded k-Center Problem. In: Chen, X., Li, B. (eds) Theory and Applications of Models of Computation. TAMC 2024. Lecture Notes in Computer Science, vol 14637. Springer, Singapore. https://doi.org/10.1007/978-981-97-2340-9_26
Download citation
DOI: https://doi.org/10.1007/978-981-97-2340-9_26
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-97-2339-3
Online ISBN: 978-981-97-2340-9
eBook Packages: Computer ScienceComputer Science (R0)