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

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