Colorful Priority $k$-Supplier

TL;DR

This paper advances approximation algorithms for the Priority $k$-Supplier problem and introduces the first algorithms for its colorful outlier variant, improving previous bounds and providing new solutions for fair clustering.

Contribution

It improves the approximation ratio for Priority $k$-Supplier with Outliers and presents the first algorithms for the Priority Colorful $k$-Supplier problem, including special cases.

Findings

Improved the approximation ratio from 9 to approximately 6.196 for Priority $k$-Supplier with Outliers.

First algorithms introduced for Priority Colorful $k$-Supplier, including a 17-approximation for general $c$ colors.

Achieved a 4.236-approximation for the case of 2 colors with fewer centers.

Abstract

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.