New algorithms for fair k-center problem with outliers and capacity constraints

The fair k-center problem has been paid lots of attention recently. In the fair k-center problem, we are given a set X of points in a metric space and a parameter k is an element of Z(+), where the points in X are divided into several groups, and each point is assigned a color to denote which group it is in. The goal is to partition X into k clusters such that the number of cluster centers with each color is equal to a given value, and the k-center problem objective is minimized. In this paper, we consider the fair k-center problem with outliers and capacity constraints, denoted as the fair k-center with outliers (FkCO) problem and the capacitated fair k-center (CFkC) problem, respectively. The outliers constraints allow up to z outliers to be discarded when computing the objective function, while the capacity constraints require that each cluster has size no more than L. In this paper, we design an Fixed-Parameter Tractability (FPT) approximation algorithm and a polynomial approximation algorithm for the above two problems. In particular, our algorithms give (1 + epsilon)-approximations with FPT time for the FkCO and CFkC problems in doubling metric space. Moreover, we also propose a 3-approximation algorithm in polynomial time for the FkCO problem with some reasonable assumptions.