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 $\varepsilon>0$, we present an improved $(16^p +\varepsilon,3)$bicriteria for this problem. Moreover, for $p=1$ ($k$median) and $p=\infty$ ($k$center), we present improved costapproximation factors $7.081+\varepsilon$ and $3+\varepsilon$ 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].
 Publication:

arXiv eprints
 Pub Date:
 June 2021
 arXiv:
 arXiv:2106.14043
 Bibcode:
 2021arXiv210614043V
 Keywords:

 Computer Science  Data Structures and Algorithms;
 Computer Science  Artificial Intelligence;
 Computer Science  Computers and Society;
 Computer Science  Machine Learning
 EPrint:
 AISTATS 2022