A Polynomial-Time Approximation for Pairwise Fair $k$-Median Clustering

TL;DR

This paper introduces the first polynomial-time approximation algorithm for pairwise fair k-median clustering with multiple groups, achieving a constant-factor approximation without violating fairness constraints.

Contribution

It presents the first polynomial-time $O(k^2 imes ext{ell} imes t)$-approximation algorithm for pairwise fair k-median clustering with more than two groups, improving upon prior bi-criteria and exponential-time methods.

Findings

Provides a polynomial-time approximation algorithm with factor $O(k^2 imes ext{ell} imes t)$

Establishes hardness results showing the problem's difficulty even for two groups

Shows the problem is nearly as hard as capacitated k-median with no known $o( ext{log} k)$ approximation

Abstract

In this work, we study pairwise fair clustering with $\ell \ge 2$ groups, where for every cluster $C$ and every group $i \in [\ell]$, the number of points in $C$ from group $i$ must be at most $t$ times the number of points in $C$ from any other group $j \in [\ell]$, for a given integer $t$. To the best of our knowledge, only bi-criteria approximation and exponential-time algorithms follow for this problem from the prior work on fair clustering problems when $\ell > 2$. In our work, focusing on the $\ell > 2$ case, we design the first polynomial-time $O(k^2\cdot \ell \cdot t)$-approximation for this problem with $k$-median cost that does not violate the fairness constraints. We complement our algorithmic result by providing hardness of approximation results, which show that our problem even when $\ell=2$ is almost as hard as the popular uniform capacitated $k$-median, for which no polynomial-time algorithm with an approximation factor of $o(\log k)$ is known.