A Subquadratic Time Approximation Algorithm for Individually Fair k-Center
Matthijs Ebbens, Nicole Funk, Jan Höckendorff, Christian Sohler, Vera Weil
TL;DR
This work advances the study of the $k$-center problem under individual fairness by presenting two bicriteria algorithms. The deterministic method achieves a $(2,2)$-approximation in $O(n^2+kn\log n)$ time, while a randomized subquadratic approach yields a $(10,2+\varepsilon)$-approximation in $O(kn\log(n/\delta)+k^2/\varepsilon)$ time (with failure probability $\delta$). A key technical advance is a subquadratic randomized procedure to approximate the fairness radii $r_k(p)$ (and related quantities) that enables efficient use of the fairness constraint without fully computing all $r_k(p)$. Together, these results push practical fair clustering closer to scalable deployment, leveraging Gonzalez-style cost-candidates and randomized radii estimates to balance fairness guarantees with running time.
Abstract
We study the $k$-center problem in the context of individual fairness. Let $P$ be a set of $n$ points in a metric space and $r_x$ be the distance between $x \in P$ and its $\lceil n/k \rceil$-th nearest neighbor. The problem asks to optimize the $k$-center objective under the constraint that, for every point $x$, there is a center within distance $r_x$. We give bicriteria $(β,γ)$-approximation algorithms that compute clusterings such that every point $x \in P$ has a center within distance $βr_x$ and the clustering cost is at most $γ$ times the optimal cost. Our main contributions are a deterministic $O(n^2+ kn \log n)$ time $(2,2)$-approximation algorithm and a randomized $O(nk\log(n/δ)+k^2/\varepsilon)$ time $(10,2+\varepsilon)$-approximation algorithm, where $δ$ denotes the failure probability. For the latter, we develop a randomized sampling procedure to compute constant factor approximations for the values $r_x$ for all $x\in P$ in subquadratic time; we believe this procedure to be of independent interest within the context of individual fairness.