Abstract:Fairness considerations have motivated new clustering problems and algorithms in recent years. In this paper we consider the Priority Matroid Median problem which generalizes the Priority $k$-Median problem that has recently been studied. The input consists of a set of facilities $\mathcal{F}$ and a set of clients $\mathcal{C}$ that lie in a metric space $(\mathcal{F} \cup \mathcal{C},d)$, and a matroid $\mathcal{M}=(\mathcal{F},\mathcal{I})$ over the facilities. In addition each client $j$ has a specified radius $r_j \ge 0$ and each facility $i \in \mathcal{F}$ has an opening cost $f_i$. The goal is to choose a subset $S \subseteq \mathcal{F}$ of facilities to minimize the $\sum_{i \in \mathcal{F}} f_i + \sum_{j \in \mathcal{C}} d(j,S)$ subject to two constraints: (i) $S$ is an independent set in $\mathcal{M}$ (that is $S \in \mathcal{I}$) and (ii) for each client $j$, its distance to an open facility is at most $r_j$ (that is, $d(j,S) \le r_j$). For this problem we describe the first bicriteria $(c_1,c_2)$ approximations for fixed constants $c_1,c_2$: the radius constraints of the clients are violated by at most a factor of $c_1$ and the objective cost is at most $c_2$ times the optimum cost. We also improve the previously known bicriteria approximation for the uniform radius setting ($r_j := L$ $\forall j \in \mathcal{C}$).
Abstract: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 were uniform, one obtains the classical $k$-center problem. Plesn\'ik [Plesn\'ik, Disc. Appl. Math. 1987] introduced this problem and gave a $2$-approximation algorithm matching the best possible algorithm for vanilla $k$-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.