On the Fixed-Parameter Tractability of Capacitated Clustering
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We study the complexity of the classic capacitated k-median and k-means problems parameterized by the number of centers, k. These problems are notoriously difficult since the best known approximation bound for high dimensional Euclidean space and general metric space is Θ(log k) and it remains a major open problem whether a constant factor exists. We show that there exists a (3+ϵ)-approximation algorithm for the capacitated k-median and a (9+ϵ)-approximation algorithm for the capacitated k-means problem in general metric spaces whose running times are f(ϵ,k) n^O(1). For Euclidean inputs of arbitrary dimension, we give a (1+ϵ)-approximation algorithm for both problems with a similar running time. This is a significant improvement over the (7+ϵ)-approximation of Adamczyk et al. for k-median in general metric spaces and the (69+ϵ)-approximation of Xu et al. for Euclidean k-means.
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