W[1]-hardness of the k -center problem parameterized by the skeleton dimension

We study the k -Center problem, where the input is a graph G=(V,E) with positive edge weights and an integer k , and the goal is to select k center vertices C ⊆ V such that the maximum distance from any vertex to the closest center vertex is minimized. In general, this problem is 𝖭𝖯 -hard and cannot be approximated within a factor less than 2. Typical applications of the k -Center problem can be found in logistics or urban planning and hence, it is natural to study the problem on transportation networks. Common characterizations of such networks are graphs that are (almost) planar or have low doubling dimension, highway dimension or skeleton dimension. It was shown by Feldmann and Marx that k -Center is 𝖶[1] -hard on planar graphs of constant doubling dimension when parameterized by the number of centers k , the highway dimension hd and the pathwidth pw (Feldmann and Marx 2020 ). We extend their result and show that even if we additionally parameterize by the skeleton dimension κ , the k -Center problem remains 𝖶[1] -hard. Moreover, we prove that under the Exponential Time Hypothesis there is no exact algorithm for k -Center that has runtime f(k,hd,pw,κ ) ·| V | ^o(pw+ κ + √(k+hd)) for any computable function f .