Abstract
In this paper, we consider the uniform capacitated k-means problem (UC-k-means), an extension of the classical k-means problem (k-means) in machine learning. In the UC-k-means, we are given a set \(\mathcal {D}\) of n points in d-dimensional space and an integer k. Every point in the d-dimensional space has an uniform capacity which is an upper bound on the number of points in \(\mathcal {D}\) that can be connected to this point. Every two-point pair in the space has an associated connecting cost, which is equal to the square of the distance between these two points. We want to find at most k points in the space as centers and connect every point in \(\mathcal {D}\) to some center without violating the capacity constraint, such that the total connecting costs is minimized. Based on the technique of local search, we present a bi-criteria approximation algorithm, which has a constant approximation guarantee and violates the cardinality constraint within a constant factor, for the UC-k-means.
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Acknowledgements
The first two authors are supported by Natural Science Foundation of China (No. 11531014). 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 Natural Science Foundation of China (No. 11871081).
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Han, L., Xu, D., Du, D. et al. An approximation algorithm for the uniform capacitated k-means problem. J Comb Optim 44, 1812–1823 (2022). https://doi.org/10.1007/s10878-020-00550-y
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DOI: https://doi.org/10.1007/s10878-020-00550-y