  In this paper we study the hardness of the $k$-Center problem on inputs that
model transportation networks. For the problem, a graph $G=(V,E)$ with edge
lengths and an integer $k$ are given and a center set $C\subseteq V$ needs to
be chosen such that $|C|\leq k$. The aim is to minimize the maximum distance of
any vertex in the graph to the closest center. This problem arises in many
applications of logistics, and thus it is natural to consider inputs that model
transportation networks. Such inputs are often assumed to be planar graphs, low
doubling metrics, or bounded highway dimension graphs. For each of these
models, parameterized approximation algorithms have been shown to exist. We
complement these results by proving that the $k$-Center problem is W[1]-hard on
planar graphs of constant doubling dimension, where the parameter is the
combination of the number of centers $k$, the highway dimension $h$, and the
pathwidth $p$. Moreover, under the Exponential Time Hypothesis there is no
$f(k,p,h)\cdot n^{o(p+\sqrt{k+h})}$ time algorithm for any computable function
$f$. Thus it is unlikely that the optimum solution to $k$-Center can be found
efficiently, even when assuming that the input graph abides to all of the above
models for transportation networks at once!
  Additionally we give a simple parameterized $(1+\varepsilon)$-approximation
algorithm for inputs of doubling dimension $d$ with runtime
$(k^k/\varepsilon^{O(kd)})\cdot n^{O(1)}$. This generalizes a previous result,
which considered inputs in $D$-dimensional $L_q$ metrics.
