Fault Tolerant Euclidean K-Centers

Abstract

The Euclidean \emph{k}-center problem is a fundamental question in computational geometry and facility location. Given a set $P$ of $n$ points in $\mathbb{R}^d$, the goal is to choose a set $F$ of $k$ center points such that the maximum distance from any point in $P$ to its nearest center in $F$ is minimized. Geometrically, this corresponds to covering all points in $P$ with $k$ balls of minimal radius. We study a natural generalization known as the $\ell$-fault-tolerant Euclidean \emph{k}-center problem, which introduces a robustness parameter $\ell \leq k$. In this variant, each point in $P$ must be covered by at least $\ell$ of the $k$ balls, or equivalently, its distance to the $\ell$th nearest center in $F$ must be minimized. This captures scenarios where redundancy is required for fault tolerance or load balancing. Our contributions include an exact $O(n \log n)$-time algorithm for solving the problem in one dimension ($\mathbb{R}$), where a linear order among points can be exploited. In two dimensions ($\mathbb{R}^2$), we prove that the problem becomes NP-hard. Nevertheless, we present an $O(nk/\ell)$-time algorithm that computes a 2-approximation, offering an efficient and practical solution with provable guarantees.February 202