Fair Representation Clustering with Several Protected Classes

Abstract

We study the problem of fair k-median where each cluster is required to have a fair representation of individuals from different groups. In the fair representation k-median problem, we are given a set of points X in a metric space. Each point x ∈ X belongs to one of ℓ groups. Further, we are given fair representation parameters αj and βj for each group j ∈ [ℓ]. We say that a k-clustering C1, • • • , C k fairly represents all groups if the number of points from group j in cluster Ci is between αj |Ci| and βj |Ci| for every j ∈ [ℓ] and i ∈ [k]. The goal is to find a set C of k centers and an assignment φ : X → C such that the clustering defined by (C, φ) fairly represents all groups and minimizes the ℓ1-objectiveWe present an O(log k)-approximation algorithm that runs in time n O(ℓ) . Note that the known algorithms for the problem either (i) violate the fairness constraints by an additive term or (ii) run in time that is exponential in both k and ℓ. We also consider an important special case of the problem where αj = βj = f j f and fj , f ∈ N for all j ∈ [ℓ]. For this special case, we present an O(log k)-approximation algorithm that runs in (kf ) O(ℓ) log n + poly(n) time.