We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP) in graphs of low highway dimension. This graph parameter roughly measures how many central nodes are visited by all shortest paths of a certain length. It has been shown that transportation networks, on which TSP and STP naturally occur for various applications in logistics, typically have a small highway dimension. While it was previously shown that these problems admit a quasi-polynomial time approximation scheme on graphs of constant highway dimension, we demonstrate that a significant improvement is possible in the special case when the highway dimension is 1. Specifically, we present a fully-polynomial time approximation scheme (FPTAS). We also prove that both TSP and STP are weakly \({\mathsf {NP}}\)-hard for these restricted graphs.

我们在低公路尺寸图中研究了旅行营业员（TSP）和斯坦纳树问题（STP）。该图参数粗略地衡量了一定长度的所有最短路径访问了多少个中央节点。已经显示出，运输网络（TSP和STP自然地出现在运输网络上，用于物流中的各种应用）通常具有较小的高速公路尺寸。虽然先前已证明这些问题允许在恒定公路尺寸的图中采用准多项式时间逼近方案，但我们证明，在高速公路尺寸为1的特殊情况下，可能会有显着改善。时间近似方案（FPTAS）。我们还证明，TSP和STP都较弱\（{\ mathsf {NP}} \）-很难处理这些受限图。