Abstract
The clustering problem has been paid lots of attention in various fields of compute science. However, in many applications, the existence of noisy data poses a big challenge for the clustering problem. As one way to deal with clustering problem with noisy data, clustering with penalties has been studied extensively, such as the k-median problem with penalties and the facility location problem with penalties. As far as we know, there is only one approximation algorithm for the k-means problem with penalties with ratio \(25+\epsilon \). All the previous related results for the clustering with penalties problems were based on the techniques of local search, LP-rounding, or primal-dual, which cannot be applied directly to the k-means problem with penalties to get better approximation ratio than \(25+\epsilon \). In this paper, we apply primal-dual technique to solve the k-means problem with penalties by a different rounding method, i.e., employing a deterministic rounding algorithm, instead of using the randomized rounding algorithm used in the previous approximation schemes. Based on the above method, an approximation algorithm with ratio \(19.849+\epsilon \) is presented for the k-means problem with penalties.
This work is supported by the National Natural Science Foundation of China under Grants (61672536, 61420106009, 61872450, 61828205), Hunan Provincial Science and Technology Program (2018WK4001).
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Feng, Q., Zhang, Z., Shi, F., Wang, J. (2019). An Improved Approximation Algorithm for the k-Means Problem with Penalties. In: Chen, Y., Deng, X., Lu, M. (eds) Frontiers in Algorithmics. FAW 2019. Lecture Notes in Computer Science(), vol 11458. Springer, Cham. https://doi.org/10.1007/978-3-030-18126-0_15
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