Revisiting Priority $k$-Center: Fairness and Outliers

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 $\max_{v \in X} \frac{1}{r(v)} d(v,S)$\mathrm{.}Ifall$r(v)${}^{\mathrm{\prime }}sareuniform,oneobtainsthe$k$-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].