The Euclidean $k$-center problem is a classical problem that has been extensively studied in computer science. Given a set $mathcal{G}$ of $n$ points in Euclidean space, the problem is to determine a set $mathcal{C}$ of $k$ centers (not necessarily part of $mathcal{G}$) such that the maximum distance between a point in $mathcal{G}$ and its nearest neighbor in $mathcal{C}$ is minimized. In this paper we study the corresponding $(k,ell)$-center problem for polygonal curves under the Frechet distance, that is, given a set $mathcal{G}$ of $n$ polygonal curves in $mathbb{R}^d$, each of complexity $m$, determine a set $mathcal{C}$ of $k$ polygonal curves in $mathbb{R}^d$, each of complexity $ell$, such that the maximum Frechet distance of a curve in $mathcal{G}$ to its closest curve in $mathcal{C}$ is minimized. In this paper, we substantially extend and improve the known approximation bounds for curves in dimension $2$ and higher. We show that, if $ell$ is part of the input, then there is no polynomial-time approximation scheme unless $mathsf{P}=mathsf{NP}$. Our constructions yield different bounds for one and two-dimensional curves and the discrete and continuous Frechet distance. In the case of the discrete Frechet distance on two-dimensional curves, we show hardness of approximation within a factor close to $2.598$. This result also holds when $k=1$, and the $mathsf{NP}$-hardness extends to the case that $ell=infty$, i.e., for the problem of computing the minimum-enclosing ball under the Frechet distance. Finally, we observe that a careful adaptation of Gonzalez algorithm in combination with a curve simplification yields a $3$-approximation in any dimension, provided that an optimal simplification can be computed exactly. We conclude that our approximation bounds are close to being tight.
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