Fixed Parameter Approximations for k-Center Problems in Low Highway Dimension Graphs

We consider the $k$-Center problem and some generalizations. For $k$-Center a set of $k$ center vertices needs to be found in a graph $G$ with edge lengths, such that the distance from any vertex of $G$ to its nearest center is minimized. This problem naturally occurs in transportation networks, and therefore we model the inputs as graphs with bounded highway dimension, as proposed by Abraham et al. [SODA 2010]. We show both approximation and fixed-parameter hardness results, and how to overcome them using fixed-parameter approximations, where the two paradigms are combined. In particular, we prove that for any $\varepsilon>0$ computing a $(2-\varepsilon)$-approximation is W[2]-hard for parameter $k$ and NP-hard for graphs with highway dimension $O(\log^2 n)$. The latter does not rule out fixed-parameter $(2-\varepsilon)$-approximations for the highway dimension parameter $h$, but implies that such an algorithm must have at least doubly exponential running time in $h$ if it exists, unless the ETH fails. On the positive side, we show how to get below the approximation factor of $2$ by combining the parameters $k$ and $h$: we develop a fixed-parameter $3/2$-approximation with running time $2^{O(kh\log h)}\cdot n^{O(1)}$. Additionally we prove that, unless P=NP, our techniques cannot be used to compute fixed-parameter $(2-\varepsilon)$-approximations for only the parameter $h$. We also provide similar fixed-parameter approximations for the weighted $k$-Center and $(k,\mathcal{F})$-Partition problems, which generalize $k$-Center.