Parameterized Approximation Algorithms for k-Center Clustering and Variants
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k-center is one of the most popular clustering models. While it admits a simple 2-approximation in polynomial time in general metrics, the Euclidean version is NP-hard to approximate within a factor of 1.93, even in the plane, if one insists the dependence on k in the running time be polynomial. Without this restriction, a classic algorithm yields a 2^O((klog k)/ϵ)dn-time (1+ϵ)-approximation for Euclidean k-center, where d is the dimension. We give a faster algorithm for small dimensions: roughly speaking an O^*(2^O((1/ϵ)^O(d)· k^1-1/d·log k))-time (1+ϵ)-approximation. In particular, the running time is roughly O^*(2^O((1/ϵ)^O(1)√(k)log k)) in the plane. We complement our algorithmic result with a matching hardness lower bound. We also consider a well-studied generalization of k-center, called Non-uniform k-center (NUkC), where we allow different radii clusters. NUkC is NP-hard to approximate within any factor, even in the Euclidean case. We design a 2^O(klog k)n^2 time 3-approximation for NUkC in general metrics, and a 2^O((klog k)/ϵ)dn time (1+ϵ)-approximation for Euclidean NUkC. The latter time bound matches the bound for k-center.
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