In the Priority $k$-Supplier problem the input consists of a metric space $(F \cup C, d)$ over set of facilities $F$ and a set of clients $C$, an integer $k > 0$, and a non-negative radius $r_v$ for each client $v \in C$. The goal is to select $k$ facilities $S \subseteq F$ to minimize $\max_{v \in C} \frac{d(v,S)}{r_v}$ where $d(v,S)$ is the distance of $v$ to the closes facility in $S$. This problem generalizes the well-studied $k$-Center and $k$-Supplier problems, and admits a $3$-approximation [Plesn\'ik, 1987, Bajpai et al., 2022. In this paper we consider two outlier versions. The Priority $k$-Supplier with Outliers problem [Bajpai et al., 2022] allows a specified number of outliers to be uncovered, and the Priority Colorful $k$-Supplier problem is a further generalization where clients are partitioned into $c$ colors and each color class allows a specified number of outliers. These problems are partly motivated by recent interest in fairness in clustering and other optimization problems involving algorithmic decision making. We build upon the work of [Bajpai et al., 2022] and improve their $9$-approximation Priority $k$-Supplier with Outliers problem to a $1+3\sqrt{3}\approx 6.196$-approximation. For the Priority Colorful $k$-Supplier problem, we present the first set of approximation algorithms. For the general case with $c$ colors, we achieve a $17$-pseudo-approximation using $k+2c-1$ centers. For the setting of $c=2$, we obtain a $7$-approximation in random polynomial time, and a $2+\sqrt{5}\approx 4.236$-pseudo-approximation using $k+1$ centers.

翻译：在优先$k$-供应商问题中，输入包括一个定义在设施集合$F$和客户集合$C$上的度量空间$(F \cup C, d)$，一个整数$k > 0$，以及每个客户$v \in C$对应的非负半径$r_v$。目标是从$F$中选择$k$个设施$S \subseteq F$，以最小化$\max_{v \in C} \frac{d(v,S)}{r_v}$，其中$d(v,S)$表示客户$v$到$S$中最近设施的距离。该问题推广了被广泛研究的$k$-中心与$k$-供应商问题，并存在一个$3$-近似算法[Plesník, 1987; Bajpai et al., 2022]。本文考虑其两种离群点变体。优先$k$-供应商带离群点问题[Bajpai et al., 2022]允许指定数量的离群点不被覆盖，而彩色优先$k$-供应商问题则进一步推广了该模型：客户被划分为$c$种颜色，每个颜色类别允许存在指定数量的离群点。这些问题的提出部分源于近期对聚类及其他涉及算法决策的优化问题中公平性研究的关注。我们在[Bajpai et al., 2022]的工作基础上，将其针对优先$k$-供应商带离群点问题的$9$-近似算法改进为$1+3\sqrt{3}\approx 6.196$-近似算法。对于彩色优先$k$-供应商问题，我们提出了首套近似算法。针对具有$c$种颜色的一般情形，我们使用$k+2c-1$个中心实现了$17$-伪近似。针对$c=2$的情形，我们在随机多项式时间内得到了$7$-近似算法，并使用$k+1$个中心实现了$2+\sqrt{5}\approx 4.236$-伪近似。