Abstract
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 \(\mathsf {NP}\)-hard for these restricted graphs. For TSP we show \(\mathsf {NP}\)-hardness for graphs of highway dimension 6, which answers an open problem posed in [Feldmann et al. ICALP 2015].
Y. Disser—Supported by the ‘Excellence Initiative’ of the German Federal and State Governments and the Graduate School CE at TU Darmstadt.
A. E. Feldmann—Supported by the Czech Science Foundation GAČR (grant #17-10090Y), and by the Center for Foundations of Modern Computer Science (Charles Univ. project UNCE/SCI/004).
M. Klimm—Supported by the German Research Foundation (DFG) as part of Math\(^+\) (project AA3-4).
J. Könemann—Supported by the Discovery Grant Program of the Natural Sciences and Engineering Research Council of Canada.
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Notes
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A metric is said to have doubling dimension d if for all \(r>0\) every ball of radius r can be covered by at most \(2^d\) balls of half the radius r/2.
- 2.
It is often assumed that all shortest paths are unique when defining the highway dimension, since this allows good polynomial approximations of this graph parameter [2]. In this work however, we do not rely on these approximations, and thus do not require uniqueness of shortest paths.
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Disser, Y., Feldmann, A.E., Klimm, M., Könemann, J. (2019). Travelling on Graphs with Small Highway Dimension. In: Sau, I., Thilikos, D. (eds) Graph-Theoretic Concepts in Computer Science. WG 2019. Lecture Notes in Computer Science(), vol 11789. Springer, Cham. https://doi.org/10.1007/978-3-030-30786-8_14
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