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Travelling on Graphs with Small Highway Dimension

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Abstract

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.

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Notes

  1. 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 [1]. In this work however, we do not rely on these approximations, and thus do not require uniqueness of shortest paths.

  3. See [26, Section 9] and [14] for detailed discussions on different definitions of the highway dimension.

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Acknowledgements

The first author is supported by the ‘Excellence Initiative’ of the German Federal and State Governments and the Graduate School CE at TU Darmstadt. The second author is supported by the Czech Science Foundation GAČR (Grant #19-27871X), and by the Center for Foundations of Modern Computer Science (Charles Univ. project UNCE/SCI/004). The third author is supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy The Berlin Mathematics Research Center MATH+ (EXC-2046/1, Project ID: 390685689). The fourth author is supported by the Discovery Grant Program of the Natural Sciences and Engineering Research Council of Canada. A preliminary version (extended abstract) of this paper appeared at WG 2019 [24]

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Authors and Affiliations

  1. Department of Mathematics, TU Darmstadt, Doliviostr. 15, 64293, Darmstadt, Germany

    Yann Disser

  2. Department of Applied Mathematics, Charles University in Prague, Malostranske nam.  25, 118 00, Praha 1, Czechia

    Andreas Emil Feldmann

  3. Institute of Mathematics, Technische Universität Berlin, Str. des 17. Juni 136, 10623, Berlin, Germany

    Max Klimm

  4. Faculty of Mathematics, University of Waterloo, 200 University Ave West, Waterloo, ON, N2L 3G1, Canada

    Jochen Könemann

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Correspondence to Andreas Emil Feldmann.

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Disser, Y., Feldmann, A.E., Klimm, M. et al. Travelling on Graphs with Small Highway Dimension. Algorithmica 83, 1352–1370 (2021). https://doi.org/10.1007/s00453-020-00785-5

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