Parameterized Approximation Algorithms for K-center Clustering and Variants

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. In this work, we give a faster algorithm for small dimensions: roughly speaking an O^*(2^{O((1/epsilon)^{O(d)} k^{1-1/d} 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, 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.

Parameterized Approximation Algorithms for Sum of Radii Clustering and Variants

“…Chakrabarty, Goyal, and Krishnaswamy (2020) shows that no constant approximation can be found for this problem in polynomial time. However, in fpt(k) time, Bandyapadhyay, Friggstad, and Mousavi (2022) showcase that the non-uniform k-center admits a simple 2approximation algorithm, offering a promising approach for this variant.…”

Red–Blue k-Center Clustering with Distance Constraints

“…Gonzalez [6] studied approximating the discrete version (centers must belong to point set). Another faster algorithm for small dimensions of the problem was introduced later by Sayan Bandyapadhyay et al [7]. Jianguang Lu et al [8] defined the uncertain constrained k-means problem and proposed a (1 + )-approximation algorithm for the problem.…”