A Subquadratic Time Approximation Algorithm for Individually Fair k-Center

TL;DR

This paper introduces subquadratic time algorithms for a fair k-center clustering problem, achieving bicriteria approximations that balance fairness constraints with clustering quality.

Contribution

It presents the first subquadratic deterministic and randomized algorithms for a fair k-center problem with bicriteria guarantees.

Findings

Deterministic $O(n^2+ kn \log n)$-time $(2,2)$-approximation algorithm.

Randomized $O(nk\log(n/\delta)+k^2/\varepsilon)$-time $(10,2+\varepsilon)$-approximation algorithm.

A novel randomized sampling method for estimating $r_x$ values efficiently.

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 $(\beta,\gamma)$-approximation algorithms that compute clusterings such that every point $x \in P$ has a center within distance $\beta r_x$ and the clustering cost is at most $\gamma$ times the optimal cost. Our main contributions are a deterministic $O(n^2+ kn \log n)$ time $(2,2)$-approximation algorithm and a randomized $O(nk\log(n/\delta)+k^2/\varepsilon)$ time $(10,2+\varepsilon)$-approximation algorithm, where $\delta$ 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.