W[1]-Hardness of the k-Center Problem Parameterized by the Skeleton Dimension

In the $k$-Center problem, we are given a graph $G=(V,E)$ with positive edge weights and an integer $k$ and the goal is to select $k$ center vertices $C \subseteq V$ such that the maximum distance from any vertex to the closest center vertex is minimized. On general graphs, the problem is NP-hard and cannot be approximated within a factor less than $2$. Typical applications of the $k$-Center problem can be found in logistics or urban planning and hence, it is natural to study the problem on transportation networks. Such networks are often characterized as graphs that are (almost) planar or have low doubling dimension, highway dimension or skeleton dimension. It was shown by Feldmann and Marx that $k$-Center is W[1]-hard on planar graphs of constant doubling dimension when parameterized by the number of centers $k$, the highway dimension $hd$ and the pathwidth $pw$. We extend their result and show that even if we additionally parameterize by the skeleton dimension $\kappa$, the $k$-Center problem remains W[1]-hard. Moreover, we prove that under the Exponential Time Hypothesis there is no exact algorithm for $k$-Center that has runtime $f(k,hd,pw,\kappa) \cdot \vert V \vert^{o(pw + \kappa + \sqrt{k+hd})}$ for any computable function $f$.