Improved Approximation Algorithms for Individually Fair Clustering
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We consider the k-clustering problem with ℓ_p-norm cost, which includes k-median, k-means and k-center cost functions, under an individual notion of fairness proposed by Jung et al. [2020]: given a set of points P of size n, a set of k centers induces a fair clustering if for every point v∈ P, v can find a center among its n/k closest neighbors. Recently, Mahabadi and Vakilian [2020] showed how to get a (p^O(p),7)-bicriteria approximation for the problem of fair k-clustering with ℓ_p-norm cost: every point finds a center within distance at most 7 times its distance to its (n/k)-th closest neighbor and the ℓ_p-norm cost of the solution is at most p^O(p) times the cost of an optimal fair solution. In this work, for any ε>0, we present an improved (16^p +ε,3)-bicriteria approximation for the fair k-clustering with ℓ_p-norm cost. To achieve our guarantees, we extend the framework of [Charikar et al., 2002, Swamy, 2016] and devise a 16^p-approximation algorithm for the facility location with ℓ_p-norm cost under matroid constraint which might be of an independent interest. Besides, our approach suggests a reduction from our individually fair clustering to a clustering with a group fairness requirement proposed by Kleindessner et al. [2019], which is essentially the median matroid problem [Krishnaswamy et al., 2011].
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