Fair Representation Clustering with Several Protected Classes
share
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 α_j and β_j for each group j∈ [ℓ]. We say that a k-clustering C_1, ⋯, C_k fairly represents all groups if the number of points from group j in cluster C_i is between α_j |C_i| and β_j |C_i| 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 ℓ_1-objective ∑_x∈ X d(x, ϕ(x)). We 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 α_j = β_j = f_j/f and f_j, 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.
READ FULL TEXT