Non-Uniform $k$-Center and Greedy Clustering

The Non-Uniform $k$-Center (NU$k$C) is a generalization of the famous $k$-center clustering problem, where we want to cover the given set of points in a metric space, using the specified number of balls of the specified radii. In the $t$-NU$k$C, we assume that the number of distinct radii is equal to $t$, and we are allowed to use $k_i$ balls of radius $r_i$, for $1 \le i \le t$. This problem was introduced by Chakrabarty et al. [ACM Trans.\ Alg.\ 16(4):46:1-46:19], who showed that a constant approximation for $t$-NU$k$C is not possible if $t$ is unbounded. On the other hand, they gave a bicriteria approximation that violates the number of allowed balls as well as the given radii by a constant factor. They also conjectured that a constant approximation for $t$-NU$k$C should be possible if $t$ is a fixed constant. Since then, there has been steady progress towards resolving this conjecture -- currently, a constant approximation for $3$-NU$k$C is known via the results of Chakrabarty and Negahbani [IPCO 2021], and Jia et al. [To appear in SOSA 2022]. We push the horizon by giving an $O(1)$-approximation for the Non-Uniform $k$-Center for $4$ distinct types of radii. Our result is obtained via a novel combination of tools and techniques from the $k$-center literature, which also demonstrates that the different generalizations of $k$-center involving non-uniform radii, and multiple coverage constraints (i.e., colorful $k$-center), are closely interlinked with each other. We hope that our ideas will contribute towards a deeper understanding of the $t$-NU$k$C problem, eventually bringing us closer to the resolution of the CGK conjecture.

翻译：美元-美元- 美元- 英特尔( NUIform $k美元) 是著名的美元- 美元- 美元- 美元- 美元- 美元。这个问题是由查克拉巴蒂等人提出的。 [ACM Trans.\ Alg.\ 16(4): 46: 46: 19], 使用指定的瑞迪球数。在美元- NU美元- 美元C, 我们假设不同的瑞迪数等于美元美元, 我们允许使用美元- 美元半径( $_ 美元) 的球, 一美元- 美元- 美元- 美元- 美元- 美元- 美元- 美元- 美元。查克拉巴蒂等人提出了这个问题。 [ACM Translickr.\ ALg.\ 16(4): 46: 46: 19], 我们想在公特尔迪尔尼特尔- 美元( 美元- 美元- 美元- 美元- 美元- 美元- 美元- 美元- 美元- 美元- 美元- 美元- 美元- 美元- 美元- 美元- 美元- 美元- 美元- 美元- 美元- 欧元- 货币- 的内基货币- 货币- 货币- 货币- 货币- 货币- 货币- 货币- 货币- 货币- 货币- 货币- 货币- 货币- 货币- 货币- 货币- 货币- 货币- 货币- 货币- 流化, 流化, 的货币- 以美元- 以以以以以以以以以以以以以以以以的货币- 以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以以