Abstract
In the classical k-means problem, we are given a data set \(\mathcal {D}\subseteq \mathbb {R}^{\ell }\) and an integer k. The object is to select a set \(S\subseteq \mathcal {R}^{\ell }\) of size at most k such that each point in \(\mathcal {D}\) is connected to the closet cluster in S with minimum total squared distances. However, in some real-life applications, it is more desirable and beneficial to pay a small penalty for not connecting some outliers in \(\mathcal {D}\) that are too far away from most points. As a result, we are motivated to study the k-means problem with penalties, for which we propose a \((6.357 + \varepsilon )\)-approximation algorithm via the primal-dual technique, improving the previous best approximation ratio of \(19.849 + \epsilon \) in [7] also by using the primal-dual technique.
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Acknowledgements
The first two authors are supported by Natural Science Foundation of China (No. 11871081). The third author is supported by the Natural Sciences and Engineering Research Council of Canada (NSERC) grant 06446, and Natural Science Foundation of China (Nos. 11771386, 11728104). The fourth author is supported by Higher Educational Science and Technology Program of Shandong Province (No. J17KA171) and Natural Science Foundation of Shandong Province (No. ZR2019MA032) of China.
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Ren, C., Xu, D., Du, D., Li, M. (2020). A Primal-Dual Algorithm for Euclidean k-Means Problem with Penalties. In: Chen, J., Feng, Q., Xu, J. (eds) Theory and Applications of Models of Computation. TAMC 2020. Lecture Notes in Computer Science(), vol 12337. Springer, Cham. https://doi.org/10.1007/978-3-030-59267-7_32
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