Improved Approximation Algorithms for Individually Fair Clustering

• URL: http://arxiv.org/abs/2106.14043v1
• Date: Sat, 26 Jun 2021 15:22:52 GMT
• Title: Improved Approximation Algorithms for Individually Fair Clustering
• Authors: Ali Vakilian, Mustafa Yal\c{c}{\i}ner
• Abstract summary: We present an improved $(16p +varepsilon,3)$-bicriteria approximation for the fair $k$-clustering with $ell_p$-norm cost. Our approach suggests a reduction from our individually fair clustering to a clustering with a group fairness requirement proposed by Kleindessner et al.
• Score: 9.914246432182873
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We consider the $k$-clustering problem with $\ell_p$-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 $v\in P$, $v$ 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 $\ell_p$-norm cost: every point finds a center within distance at most $7$ times its distance to its $(n/k)$-th closest neighbor and the $\ell_p$-norm cost of the solution is at most $p^{O(p)}$ times the cost of an optimal fair solution. In this work, for any $\varepsilon>0$, we present an improved $(16^p +\varepsilon,3)$-bicriteria approximation for the fair $k$-clustering with $\ell_p$-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 $\ell_p$-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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