Abstract
In this paper, we study the k-means problem with (nonuniform) penalties (k-MPWP) which is a natural generalization of the classic k-means problem. In the k-MPWP, we are given an n-client set \( {\mathcal {D}} \subset {\mathbb {R}}^d\), a penalty cost \(p_j>0\) for each \(j \in {\mathcal {D}}\), and an integer \(k \le n\). The goal is to open a center subset \(F \subset {\mathbb {R}}^d\) with \( |F| \le k\) and to choose a client subset \(P \subseteq {\mathcal {D}} \) as the penalized client set such that the total cost (including the sum of squares of distance for each client in \( {\mathcal {D}} \backslash P \) to the nearest open center and the sum of penalty cost for each client in P) is minimized. We offer a local search \(( 81+ \varepsilon )\)-approximation algorithm for the k-MPWP by using single-swap operation. We further improve the above approximation ratio to \(( 25+ \varepsilon )\) by using multi-swap operation.
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Acknowledgements
The research of the first author is supported by Higher Educational Science and Technology Program of Shandong Province (No. J15LN23) and the Science and Technology Development Plan Project of Jinan City (No. 201401211). The second author is supported by Ri-Xin Talents Project of Beijing University of Technology. The third author is supported by Natural Science Foundation of China (No. 11501412). The fourth author is supported by Natural Science Foundation of China (No. 11531014). The fifth author is supported by Beijing Excellent Talents Funding (No. 2014000020124G046).
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A preliminary version of this paper appeared in Proceedings of the 23rd International Computing and Combinatorics Conference, pp. 568–574, 2017.
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Zhang, D., Hao, C., Wu, C. et al. Local search approximation algorithms for the k-means problem with penalties. J Comb Optim 37, 439–453 (2019). https://doi.org/10.1007/s10878-018-0278-6
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DOI: https://doi.org/10.1007/s10878-018-0278-6