The Fair k-Center with Outliers Problem: FPT and Polynomial Approximations

Xiaoliang Wu, Qilong Feng, Jinhui Xu, Jianxin Wang

Abstract: The fair k-center and k-center with outliers problems are two important variants of the k-center problem in computer science, which have attracted lots of attention. The previous best results for the above two problems are a 3-approximation (ICML 2020) and a 2-approximation (ICALP 2016), respectively. In this paper, we consider a common generalization of the two mentioned variants of the k-center problem, denoted as the fair k-center with outliers (FkCO) problem. For the FkCO problem, we are given a set X of points in a metric space and parameters k and z, where the points in X are divided into several groups, and each point is assigned a color to denote which group it belongs to. The goal is to find a subset \(C\subseteq X\) of k centers and a set Z of at most z outliers such that C satisfies fairness constraints, and the objective \(\max _{x\in X{\setminus Z}}\min _{c\in C}d(x, c)\) is minimized. In this paper, we study the Fixed-Parameter Tractability (FPT) approximation algorithm and polynomial approximation algorithm for the FkCO problem. Our main contributions are: (1) we propose a \((1+\epsilon )\)-approximation algorithm in FPT time for the FkCO problem in a low-dimensional doubling metric space; (2) we achieve a polynomial 3-approximation algorithm for the FkCO problem with the reasonable assumptions that all optimal clusters are well separated and have size greater than z.