Improved Approximation Algorithms for Individually Fair Clustering

Ali Vakilian, Mustafa Yalciner
Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, PMLR 151:8758-8779, 2022.

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.

Cite this Paper

BibTeX
@InProceedings{pmlr-v151-vakilian22a, title = { Improved Approximation Algorithms for Individually Fair Clustering }, author = {Vakilian, Ali and Yalciner, Mustafa}, booktitle = {Proceedings of The 25th International Conference on Artificial Intelligence and Statistics}, pages = {8758--8779}, year = {2022}, editor = {Camps-Valls, Gustau and Ruiz, Francisco J. R. and Valera, Isabel}, volume = {151}, series = {Proceedings of Machine Learning Research}, month = {28--30 Mar}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v151/vakilian22a/vakilian22a.pdf}, url = {https://proceedings.mlr.press/v151/vakilian22a.html}, 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. } }
Endnote
%0 Conference Paper %T Improved Approximation Algorithms for Individually Fair Clustering %A Ali Vakilian %A Mustafa Yalciner %B Proceedings of The 25th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2022 %E Gustau Camps-Valls %E Francisco J. R. Ruiz %E Isabel Valera %F pmlr-v151-vakilian22a %I PMLR %P 8758--8779 %U https://proceedings.mlr.press/v151/vakilian22a.html %V 151 %X 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.
APA
Vakilian, A. & Yalciner, M.. (2022). Improved Approximation Algorithms for Individually Fair Clustering . Proceedings of The 25th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 151:8758-8779 Available from https://proceedings.mlr.press/v151/vakilian22a.html.

Related Material