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doi: 10.3934/jimo.2021067
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## Approximation algorithm for spherical $k$-means problem with penalty

 1 School of Business, Nankai University, Tianjin, 300071, China, College of Science, Tianjin University of Technology, Tianjin, 300384, China 2 Computer Science and Technology Department, Tianjin Renai College, Tianjin, 306136, China 3 Department of Operations Research and Information Engineering, Beijing University of Technology, Beijing, 100124, China

* Corresponding author: Wei Lv

Received  July 2020 Revised  February 2021 Early access April 2021

The $k$-means problem is a classical combinatorial optimization problem which has lots of applications in many fields such as machine learning, data mining, etc. We consider a variant of $k$-means problem in the spherical space, that is, spherical $k$-means problem with penalties. In the problem, it is allowable that some nodes in the spherical space can not be clustered by paying some penalty costs. Based on local search scheme, we propose a $\left(4 (11+4\sqrt{7})+ \epsilon\right)$-approximation algorithm using singe-swap operation, where $\epsilon$ is a positive constant.

Citation: Chenchen Wu, Wei Lv, Yujie Wang, Dachuan Xu. Approximation algorithm for spherical $k$-means problem with penalty. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021067
##### References:
 [1] S. Ahmadian, A. Norouzi-Fard, O. Svensson and J. Ward, Better guarantees for $k$-means and Euclidean $k$-median by primal-dual algorithms, SIAM Journal on Computing, 49 (2019), FOCS17-97–FOCS17-156. doi: 10.1137/18M1171321.  Google Scholar [2] D. Aloise, A. Deshpande, P. Hansen and P. Popat, NP-hardness of Euclidean sum-of-squares clustering, Machine Learning, 75 (2009), 245-248.  doi: 10.1007/s10994-009-5103-0.  Google Scholar [3] S. Ahmadian, A. Norouzi-Fard, O. Svensson and J. Ward, Better guarantees for $k$-means and Euclidean $k$-median by primal-dual algorithms, SIAM Journal on Computing, 49 (2019), FOCS17-97–FOCS17-156. doi: 10.1137/18M1171321.  Google Scholar [4] D. Arthur and S. Vassilvitskii, $K$-means++: The advantages of careful seeding, Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2007, 1027–1035. doi: 10.1145/1283383.1283494.  Google Scholar [5] Y. Endo and S. Miyamoto, Spherical $k$-means++ clustering, Proceedings of International Conference on Modeling Decisions for Artificial Intelligence, (2015), 103–114. doi: 10.1007/978-3-319-23240-99.  Google Scholar [6] Q. Feng, Z. Zhang, F. Shi and J. Wang, An improved approximation algorithm for the $k$-means problem with penalties, Proceedings of International Workshop on Frontiers in Algorithmics, (2019), 170–181. doi: 10.1007/978-3-030-18126-015.  Google Scholar [7] Z. Friggstad, K. Khodamoradi, M. Rezapour and M. Salavatipour, Approximation schemes for clustering with outliers, ACM Transactions on Algorithms, 15 (2019), 26: 1–26: 26. doi: 10.1145/3301446.  Google Scholar [8] A. Georgogiannis, Robust $k$-means: A theoretical revisit, Proceedings of 30th Conference on Neural Information Processing Systems, (2016), 2891–2899. doi: 10.5555/3157382.3157421.  Google Scholar [9] J. Han, M. Kamber and J. Pei, Data Mining: Concepts and Techniques, Elsevier, 2012. doi: 10.1016/C2009-0-61819-5.  Google Scholar [10] A. K. Jain, Data clustering: 50 years beyond $k$-means, Pattern Recognition Letters, 31 (2010), 651-666.  doi: 10.1016/j.patrec.2009.09.011.  Google Scholar [11] S. Ji, D. Xu, L. Guo, M. Li and D. Zhang, The seeding algorithm for spherical $k$-means clustering with penalties, Journal of Combinatorial Optimization, 2 (2020), 1-18.  doi: 10.1007/s10898-019-00779-w.  Google Scholar [12] T. Kanungo, D. M. Mount, N. S. Netanyahu, C. D. Piatko, R. Silverman and A. Y. Wu, A local search approximation algorithm for $k$-means clustering, Computational Geometry-Theory and Applications, 28 (2004), 89-112.  doi: 10.1016/j.comgeo.2004.03.003.  Google Scholar [13] M. Li, The bi-criteria seeding algorithms for two variants of $k$-means problem, Journal of Combinatorial Optimization. doi: 10.1007/s10878-020-00537-9.  Google Scholar [14] M. Li, D. Xu, J. Yue, D. Zhang and P. Zhang, The seeding algorithm for $k$-means problem with penalties, Journal of Combinatorial Optimization, 39 (2020), 15-32.  doi: 10.1007/s10878-019-00450-w.  Google Scholar [15] S. Lloyd, Least squares quantization in PCM, IEEE Transactions on Information Theory, 28 (1982), 129-137.  doi: 10.1109/TIT.1982.1056489.  Google Scholar [16] R. Ostrovsky, Y. Rabani, L. Schulman and C. Swamy, The effectiveness of Lloyd-type methods for the $k$-means problem, Journal of the ACM, 59 (2012), 28: 1–28: 22. doi: 10.1145/2395116.2395117.  Google Scholar [17] D. Wei, A constant-factor bi-criteria approximation guarantee for $k$-means++, Proceedings of the Thirtieth International Conference on Neural Information Processing Systems, (2016), 604–612. doi: 10.5555/3157096.3157164.  Google Scholar [18] X. Wu, V. Kumar, J. Quinlan, J. Ross Ghosh, Q. Yang, H. Motoda, G. J. McLachlan, A. Ng, B. Liu, P. S. Yu, Z. H. Zhou, M. Steinbach, D. J. Hand and D. Steinberg, Top 10 algorithms in data mining, Knowledge and Information Systems, 14 (2008), 1-37.  doi: 10.1007/s10115-007-0114-2.  Google Scholar [19] D. Zhang, Y. Cheng, M. Li, Y. Wang and D. Xu, Local search approximation algorithms for the spherical $k$-means problem, Proceedings of International Conference on Algorithmic Applications in Management, Springer (2019), 341–351. doi: 10.1007/978-3-030-27195-4_31.  Google Scholar [20] D. Zhang, C. Hao, C. Wu, D. Xu and Z. Zhang, Local search approximation algorithms for the $k$-means problem with penalties, Journal of Combinatorial Optimization, 37 (2019), 439-453.  doi: 10.1007/s10878-018-0278-6.  Google Scholar

show all references

##### References:
 [1] S. Ahmadian, A. Norouzi-Fard, O. Svensson and J. Ward, Better guarantees for $k$-means and Euclidean $k$-median by primal-dual algorithms, SIAM Journal on Computing, 49 (2019), FOCS17-97–FOCS17-156. doi: 10.1137/18M1171321.  Google Scholar [2] D. Aloise, A. Deshpande, P. Hansen and P. Popat, NP-hardness of Euclidean sum-of-squares clustering, Machine Learning, 75 (2009), 245-248.  doi: 10.1007/s10994-009-5103-0.  Google Scholar [3] S. Ahmadian, A. Norouzi-Fard, O. Svensson and J. Ward, Better guarantees for $k$-means and Euclidean $k$-median by primal-dual algorithms, SIAM Journal on Computing, 49 (2019), FOCS17-97–FOCS17-156. doi: 10.1137/18M1171321.  Google Scholar [4] D. Arthur and S. Vassilvitskii, $K$-means++: The advantages of careful seeding, Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SIAM, 2007, 1027–1035. doi: 10.1145/1283383.1283494.  Google Scholar [5] Y. Endo and S. Miyamoto, Spherical $k$-means++ clustering, Proceedings of International Conference on Modeling Decisions for Artificial Intelligence, (2015), 103–114. doi: 10.1007/978-3-319-23240-99.  Google Scholar [6] Q. Feng, Z. Zhang, F. Shi and J. Wang, An improved approximation algorithm for the $k$-means problem with penalties, Proceedings of International Workshop on Frontiers in Algorithmics, (2019), 170–181. doi: 10.1007/978-3-030-18126-015.  Google Scholar [7] Z. Friggstad, K. Khodamoradi, M. Rezapour and M. Salavatipour, Approximation schemes for clustering with outliers, ACM Transactions on Algorithms, 15 (2019), 26: 1–26: 26. doi: 10.1145/3301446.  Google Scholar [8] A. Georgogiannis, Robust $k$-means: A theoretical revisit, Proceedings of 30th Conference on Neural Information Processing Systems, (2016), 2891–2899. doi: 10.5555/3157382.3157421.  Google Scholar [9] J. Han, M. Kamber and J. Pei, Data Mining: Concepts and Techniques, Elsevier, 2012. doi: 10.1016/C2009-0-61819-5.  Google Scholar [10] A. K. Jain, Data clustering: 50 years beyond $k$-means, Pattern Recognition Letters, 31 (2010), 651-666.  doi: 10.1016/j.patrec.2009.09.011.  Google Scholar [11] S. Ji, D. Xu, L. Guo, M. Li and D. Zhang, The seeding algorithm for spherical $k$-means clustering with penalties, Journal of Combinatorial Optimization, 2 (2020), 1-18.  doi: 10.1007/s10898-019-00779-w.  Google Scholar [12] T. Kanungo, D. M. Mount, N. S. Netanyahu, C. D. Piatko, R. Silverman and A. Y. Wu, A local search approximation algorithm for $k$-means clustering, Computational Geometry-Theory and Applications, 28 (2004), 89-112.  doi: 10.1016/j.comgeo.2004.03.003.  Google Scholar [13] M. Li, The bi-criteria seeding algorithms for two variants of $k$-means problem, Journal of Combinatorial Optimization. doi: 10.1007/s10878-020-00537-9.  Google Scholar [14] M. Li, D. Xu, J. Yue, D. Zhang and P. Zhang, The seeding algorithm for $k$-means problem with penalties, Journal of Combinatorial Optimization, 39 (2020), 15-32.  doi: 10.1007/s10878-019-00450-w.  Google Scholar [15] S. Lloyd, Least squares quantization in PCM, IEEE Transactions on Information Theory, 28 (1982), 129-137.  doi: 10.1109/TIT.1982.1056489.  Google Scholar [16] R. Ostrovsky, Y. Rabani, L. Schulman and C. Swamy, The effectiveness of Lloyd-type methods for the $k$-means problem, Journal of the ACM, 59 (2012), 28: 1–28: 22. doi: 10.1145/2395116.2395117.  Google Scholar [17] D. Wei, A constant-factor bi-criteria approximation guarantee for $k$-means++, Proceedings of the Thirtieth International Conference on Neural Information Processing Systems, (2016), 604–612. doi: 10.5555/3157096.3157164.  Google Scholar [18] X. Wu, V. Kumar, J. Quinlan, J. Ross Ghosh, Q. Yang, H. Motoda, G. J. McLachlan, A. Ng, B. Liu, P. S. Yu, Z. H. Zhou, M. Steinbach, D. J. Hand and D. Steinberg, Top 10 algorithms in data mining, Knowledge and Information Systems, 14 (2008), 1-37.  doi: 10.1007/s10115-007-0114-2.  Google Scholar [19] D. Zhang, Y. Cheng, M. Li, Y. Wang and D. Xu, Local search approximation algorithms for the spherical $k$-means problem, Proceedings of International Conference on Algorithmic Applications in Management, Springer (2019), 341–351. doi: 10.1007/978-3-030-27195-4_31.  Google Scholar [20] D. Zhang, C. Hao, C. Wu, D. Xu and Z. Zhang, Local search approximation algorithms for the $k$-means problem with penalties, Journal of Combinatorial Optimization, 37 (2019), 439-453.  doi: 10.1007/s10878-018-0278-6.  Google Scholar
$k$-means problem
Spherical $k$-means problem
Swap operataion
Compound mapping
Cases for re-clustering
Partition of $\mathcal{N}$
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