Tight FPT Approximation for Socially Fair Clustering

• Published in: Information processing letters Vol. 182; p. 106383
• Main Authors: Goyal, Dishant, Jaiswal, Ragesh
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.08.2023
• Subjects: Approximation algorithms; Clustering; Fairness; Fixed-parameter tractability; k-means; k-means; Approximation algorithms; Clustering; Fixed-parameter tractability; Fairness
• ISSN: 0020-0190; 1872-6119
• DOI: 10.1016/j.ipl.2023.106383

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 X with a distance function d(.,.). There are ℓ groups: P1,…,Pℓ⊆P. We are also given a set F of feasible centers in X. The goal in the socially fair k-median problem is to find a set C⊆F of k centers that minimizes the maximum average cost over all the groups. That is, find C that minimizes the objective function Φ(C,P)≡maxj⁡{∑x∈Pjd(C,x)/|Pj|}, 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., d2(.,.) instead of d(.,.). The current best approximation guarantee for both of the problems is O(log⁡ℓlog⁡log⁡ℓ) due to Makarychev and Vakilian (COLT 2021). In this work, we study the fixed-parameter tractability of the problems with respect to parameter k. We design (3+ε) and (9+ε) approximation algorithms for the socially fair k-median and k-means problems, respectively, in FPT (fixed-parameter tractable) time f(k,ε)⋅nO(1), where f(k,ε)=(k/ε)O(k) and n=|P∪F|. The algorithms are randomized and succeed with a probability of at least (1−1n). Furthermore, we show that if W[2]≠FPT, then better approximation guarantees are not possible in FPT time. •We design an FPT time approximation algorithm for the socially fair clustering problem.•We design a polynomial time bi-criteria approximation algorithm that gives (1+epsilon)-approximation for the problem.•We show that the bi-criteria approximation algorithm can be converted to FPT time approximation algorithm.•We show the matching lower bounds for the socially fair clustering problem.