Colorful Priority \(k\)-Supplier

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 [Plesník, 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.