FPT Constant-Approximations for Capacitated Clustering to Minimize the Sum of Cluster Radii
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Clustering with capacity constraints is a fundamental problem that attracted significant attention throughout the years. In this paper, we give the first FPT constant-factor approximation algorithm for the problem of clustering points in a general metric into k clusters to minimize the sum of cluster radii, subject to non-uniform hard capacity constraints. In particular, we give a (15+ϵ)-approximation algorithm that runs in 2^0(k^2log k)· n^3 time. When capacities are uniform, we obtain the following improved approximation bounds: A (4 + ϵ)-approximation with running time 2^O(klog(k/ϵ))n^3, which significantly improves over the FPT 28-approximation of Inamdar and Varadarajan [ESA 2020]; a (2 + ϵ)-approximation with running time 2^O(k/ϵ^2 ·log(k/ϵ))dn^3 and a (1+ϵ)-approximation with running time 2^O(kdlog ((k/ϵ)))n^3 in the Euclidean space; and a (1 + ϵ)-approximation in the Euclidean space with running time 2^O(k/ϵ^2 ·log(k/ϵ))dn^3 if we are allowed to violate the capacities by (1 + ϵ)-factor. We complement this result by showing that there is no (1 + ϵ)-approximation algorithm running in time f(k)· n^O(1), if any capacity violation is not allowed.
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