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- W4225839904 abstract "$k$-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 $2^{O((klog k)/{epsilon})}dn$-time $(1+epsilon)$-approximation for Euclidean $k$-center, where $d$ is the dimension. We give a faster algorithm for small dimensions: roughly speaking an $O^*(2^{O((1/epsilon)^{O(d)} cdot k^{1-1/d} cdot log k)})$-time $(1+epsilon)$-approximation. In particular, the running time is roughly $O^*(2^{O((1/epsilon)^{O(1)}sqrt{k}log k)})$ 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 $2^{O((klog k)/epsilon)}dn$ time $(1+epsilon)$-approximation for Euclidean NUkC. The latter time bound matches the bound for $k$-center." @default.
- W4225839904 created "2022-05-05" @default.
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- W4225839904 date "2021-12-19" @default.
- W4225839904 modified "2023-09-28" @default.
- W4225839904 title "Parameterized Approximation Algorithms for $k$-Center Clustering and Variants" @default.
- W4225839904 doi "https://doi.org/10.48550/arxiv.2112.10195" @default.
- W4225839904 hasPublicationYear "2021" @default.
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