Approximation Algorithms for Fair Range Clustering
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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 = P_1 ⊎ P_2 ⊎⋯⊎ P_ℓ) and a set of ℓ intervals [α_1, β_1], ⋯, [α_ℓ, β_ℓ] on desired number of centers from each group, the goal is to pick a set of k centers C with minimum ℓ_p-clustering cost (i.e., (∑_v∈ P d(v,C)^p)^1/p) such that for each group i∈ℓ, |C∩ P_i| ∈ [α_i, β_i]. In particular, the fair range ℓ_p-clustering captures fair range k-center, k-median and k-means as its special cases. In this work, we provide efficient constant factor approximation algorithms for fair range ℓ_p-clustering for all values of p∈ [1,∞).
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