# FPT Approximation for Socially Fair Clustering

In this work, we study the socially fair k-median/k-means problem. We are given a set of points P in a metric space π³ with a distance function d(.,.). There are β groups: P_1,β¦,P_ββ P. We are also given a set F of feasible centers in π³. The goal of the socially fair k-median problem is to find a set C β F of k centers that minimizes the maximum average cost over all the groups. That is, find C that minimizes the objective function Ξ¦(C,P) β‘max_jβ_x β P_j d(C,x)/|P_j|, where d(C,x) is the distance of x to the closest center in C. The socially fair k-means problem is defined similarly by using squared distances, i.e., d^2(.,.) instead of d(.,.). In this work, we design (5+Ξ΅) and (33 + Ξ΅) approximation algorithms for the socially fair k-median and k-means problems, respectively. For the parameters: k and β, the algorithms have an FPT (fixed parameter tractable) running time of f(k,β,Ξ΅) Β· n for f(k,β,Ξ΅) = 2^O(k β/Ξ΅) and n = |P βͺ F|. We also study a special case of the problem where the centers are allowed to be chosen from the point set P, i.e., P β F. For this special case, our algorithms give better approximation guarantees of (4+Ξ΅) and (18+Ξ΅) for the socially fair k-median and k-means problems, respectively. Furthermore, we convert these algorithms to constant pass log-space streaming algorithms. Lastly, we show FPT hardness of approximation results for the problem with a small gap between our upper and lower bounds.

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