In the Priority $k$-Center problem, the input consists of a metric space $(X,d)$, an integer $k$, and for each point $v \in X$ a priority radius $r(v)$. The goal is to choose $k$-centers $S \subseteq X$ to minimize $\max_{v \in X} \frac{1}{r(v)} d(v,S)$. If all $r(v)$'s are uniform, one obtains the $k$-Center problem. Plesn\'ik [Plesn\'ik, Disc. Appl. Math. 1987] introduced the Priority $k$-Center problem and gave a $2$-approximation algorithm matching the best possible algorithm for $k$-Center. We show how the problem is related to two different notions of fair clustering [Harris et al., NeurIPS 2018; Jung et al., FORC 2020]. Motivated by these developments we revisit the problem and, in our main technical contribution, develop a framework that yields constant factor approximation algorithms for Priority $k$-Center with outliers. Our framework extends to generalizations of Priority $k$-Center to matroid and knapsack constraints, and as a corollary, also yields algorithms with fairness guarantees in the lottery model of Harris et al [Harris et al, JMLR 2019].
翻译:在Rior $k$- center 问题中,投入包括一个公吨空间 $(X,d) 美元,一个整数美元,对于每个点,一个整数美元;对于每个点,一个优先半径 $(v) 美元。目标是选择美元-center $S subseteq $S subseteq $S subseteq $S subsetequal $S $S subseteq $S $S subseteqlority $(xxx) d(v),d(v),S。如果所有r(v) 美元是统一的,一个获得美元- centerral 问题。Plesn\'ik [Plesnralialiik, Disc. Appl. Math. 1987] 引入了美元-Central 问题,并给出了一个符合美元最高比率框架的$C, 并且将一个固定比率框架, 发展了一个我们最高级的美元-ralexalexilationalations 。