Tight 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 metric space X with distance function d(.,.). There ℓ groups: P1,…,Pℓ⊆P. also F feasible centers X. The goal k-median problem is to find C⊆F k that minimizes maximum average cost over all groups. That is, C objective Φ(C,P)≡maxj⁡{∑x∈Pjd(C,x)/|Pj|}, where d(C,x) x closest center C. k-means defined similarly by using squared distances, i.e., d2(.,.) instead current best approximation guarantee for both problems O(log⁡ℓlog⁡log⁡ℓ) due Makarychev and Vakilian (COLT 2021). fixed-parameter tractability respect parameter k. design (3+ε) (9+ε) algorithms problems, respectively, FPT (fixed-parameter tractable) time f(k,ε)⋅nO(1), f(k,ε)=(k/ε)O(k) n=|P∪F|. randomized succeed probability at least (1−1n). Furthermore, show if W[2]≠FPT, then better guarantees not possible time.