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Travelling on Graphs with Small Highway Dimension

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Book cover Graph-Theoretic Concepts in Computer Science (WG 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11789))

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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 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

  1. 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. 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.

  3. 3.

    The proofs of Theorems 3 and 4 are deferred to the full version of the paper.

  4. 4.

    See [24, Section 9] and [15] for detailed discussions on different definitions of the highway dimension.

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Authors and Affiliations

  1. TU Darmstadt, Darmstadt, Germany

    Yann Disser

  2. Charles University in Prague, Prague, Czechia

    Andreas Emil Feldmann

  3. Humboldt-Universität zu Berlin, Berlin, Germany

    Max Klimm

  4. University of Waterloo, Waterloo, Canada

    Jochen Könemann

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  1. Yann Disser
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  2. Andreas Emil Feldmann
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Correspondence to Andreas Emil Feldmann .

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Editors and Affiliations

  1. CNRS, LIRMM, Université de Montpellier, Montpellier, France

    Ignasi Sau

  2. CNRS, LIRMM, Université de Montpellier, Montpellier, France

    Dimitrios M. Thilikos

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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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  • DOI: https://doi.org/10.1007/978-3-030-30786-8_14

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