  We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP)
in graphs of low highway dimension. This graph parameter was introduced by
Abraham et al. [SODA 2010] as a model for transportation networks, on which TSP
and STP naturally occur for various applications in logistics. It was
previously shown [Feldmann et al. ICALP 2015] that these problems admit a
quasi-polynomial time approximation scheme (QPTAS) on graphs of constant
highway dimension. We demonstrate that a significant improvement is possible in
the special case when the highway dimension is 1, for which we present a
fully-polynomial time approximation scheme (FPTAS). We also prove that STP is
weakly NP-hard for these restricted graphs. For TSP we show NP-hardness for
graphs of highway dimension 6, which answers an open problem posed in [Feldmann
et al. ICALP 2015].
