Generalized k-center: distinguishing doubling and highway dimension

We consider generalizations of the k -CENTER problem in graphs of low doubling and highway dimension. For the CAPACITATED k -SUPPLIER WITH OUTLIERS (CKSWO) problem, we show an efficient parameterized approximation scheme (EPAS) when the parameters are k, the number of outliers and the doubling dimension of the supplier set. On the other hand, we show that for the CAPACITATED k -CENTER problem, which is a special case of CKSWO, obtaining a parameterized approximation scheme (PAS) is W[1]-hard when the parameters are k, and the highway dimension. This is the first known example of a problem for which it is hard to obtain a PAS for highway dimension, while simultaneously admitting an EPAS for doubling dimension.