Approximation Algorithms for Fair Range Clustering

This paper studies the fair range clustering problem in which the data points are from different demographic groups and the goal is to pick k centers with the minimum clustering cost such that each group is at least minimally represented in the centers set and no group dominates the centers set. More precisely, given a set of n points in a metric space (P, d) where each point belongs to one of the ℓ different demographics (i.e., P = P1 ⊎ P2 ⊎ · · · ⊎ Pℓ) and a set of ℓ intervals [α1, β1], · · · , [αℓ, βℓ] on desired number of centers from each group, the goal is to pick a set of k centers C with minimum ℓp-clustering cost such that for each group i ∈ ℓ, |C ∩ Pi| ∈ [αi, βi]. In particular, the fair range ℓp-clustering captures fair range k-center, k-median and k-means as its special cases. In this work, we provide an efficient constant factor approximation algorithm for the fair range ℓp-clustering for all values of p ∈ [1, ∞).

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