Improved Approximation Algorithms for Individually Fair Clustering

Abstract

We consider the $k$-clustering problem with ${\ell }_p$-norm cost, which includes $k$-median, $k$-means and $k$-center, 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 every point in $P$ has a center among its $n/k$ closest neighbors. Mahabadi and Vakilian [2020] presented a $(p^{O(p)},7)$-bicriteria approximation for fair 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 $\epsilon >0$, we present an improved $(16^p+\epsilon ,3)$-bicriteria for this problem. Moreover, for $p=1$ ($k$-median) and $p=\infty$ ($k$-center), we present improved cost-approximation factors $7.081+\epsilon$ and $3+\epsilon$ respectively. 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].

arXiv commentAISTATS 2022

References in corpus (11)

• The Frontiers of Fairness in Machine Learning
• Auditing for Discrimination in Algorithms Delivering Job Ads
• Clustering without Over-Representation
• Scalable Fair Clustering
• Coresets for Clustering with Fairness Constraints
• Proportionally Fair Clustering
• Individual Fairness for $k$-Clustering
• A Notion of Individual Fairness for Clustering
• Distributional Individual Fairness in Clustering
• A Center in Your Neighborhood: Fairness in Facility Location
• Approximation Algorithms for Socially Fair Clustering

Cited by in corpus (2)

• Near-Optimal Explainable $k$-Means for All Dimensions
• Spectral Normalized-Cut Graph Partitioning with Fairness Constraints