Generalized $k$-Center: Distinguishing Doubling and Highway Dimension

TL;DR

This paper explores generalized $k$-Center problems in graphs with low doubling and highway dimensions, providing efficient algorithms for some cases and hardness results for others, highlighting differences between these graph parameters.

Contribution

It introduces an EPAS for Capacitated $k$-Supplier with Outliers in low doubling dimension graphs and proves W[1]-hardness for Capacitated $k$-Center with respect to highway dimension.

Findings

EPAS for CkSwO in low doubling dimension graphs

W[1]-hardness for Capacitated $k$-Center with highway dimension

First example of hardness difference between doubling and highway dimensions

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.