Abstrakt
In this paper we study the hardness of the 𝑘-CENTER problem on inputs that model transportation networks. For the problem, a graph 𝐺=(𝑉,��) with edge lengths and an integer k are given and a center set 𝐶 needs to be chosen such that |𝐶|<=𝑘.
The aim is to minimize the maximum distance of any vertex in the graph to the closest center. This problem arises in many applications of logistics, and thus it is natural to consider inputs that model transportation networks.
Such inputs are often assumed to be planar graphs, low doubling metrics, or bounded highway dimension graphs. For each of these models, parameterized approximation algorithms have been shown to exist.
We complement these results by proving that the 𝑘-CENTER problem is W[1]-hard on planar graphs of constant doubling dimension, where the parameter is the combination of the number of centers k, the highway dimension h, and the pathwidth p.