Abstract
The k-means problem is a classic NP-hard problem in machine learning and computational geometry. And its goal is to separate the given set into k clusters according to the minimal squared distance. The k-means problem with penalties, as one generalization of k-means problem, allows that some point need not be clustered instead of being paid some penalty. In this paper, we study the k-means problem with penalties by using the seeding algorithm. We propose that the accuracy only involves the ratio of the maximal penalty value to the minimal one. When the penalty is uniform, the approximation factor reduces to the same one for the k-means problem. Moreover, our result generalizes the k-means++ for k-means problem to the penalty version. Numerical experiments show that our seeding algorithm is more effective than the one without using seeding.
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Acknowledgements
The research of the first author is supported by Higher Educational Science and Technology Program of Shandong Province (No. J17KA171). The second author is supported by National Natural Science Foundation of China (No. 11531014). The third author is supported by National Natural Science Foundation of China (Nos. 11626148 and 11701342) and Natural Science Foundation of Shandong Province (No. ZR2016AQ01). The fourth author’s research is supported by National Natural Science Foundation of China (No. 11871081). The fifth author is supported by National Natural Science Foundation of China (Nos. 61672323 and 61972228).
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Li, M., Xu, D., Yue, J. et al. The seeding algorithm for k-means problem with penalties. J Comb Optim 39, 15–32 (2020). https://doi.org/10.1007/s10878-019-00450-w
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DOI: https://doi.org/10.1007/s10878-019-00450-w