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 ε > 0 computing a (2− ε)-approximation is W[2]-hard for parameter k, and NP-hard for graphs with highway dimension O(log n). The latter does not rule out fixed-parameter (2− ε)-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 log h) · n. Additionally we prove that, unless P=NP, our techniques cannot be used to compute fixed-parameter (2− ε)-approximations for only the parameter h. We also provide similar fixed-parameter approximations for the weighted k-Center and (k,F)-Partition problems, which generalize k-Center.