Generalized k-Center: Distinguishing Doubling and Highway Dimension

Abstract

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 $$\mathrm {W[1]}$$ -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.