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Problems Hard for Treewidth but Easy for Stable Gonality

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Graph-Theoretic Concepts in Computer Science (WG 2022)

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Abstract

We show that some natural problems that are XNLP-hard (hence \(\textrm{W}[t]\)-hard for all t) when parameterized by pathwidth or treewidth, become FPT when parameterized by stable gonality, a novel graph parameter based on optimal maps from graphs to trees. The problems we consider are classical flow and orientation problems, such as Undirected Flow with Lower Bounds, Minimum Maximum Outdegree, and capacitated optimization problems such as Capacitated (Red-Blue) Dominating Set. Our hardness claims beat existing results. The FPT algorithms use a new parameter “treebreadth”, associated to a weighted tree partition, as well as DP and ILP.

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Notes

  1. 1.

    In [23] it is required that \(\textrm{val}(f) \ge R\) rather than \(\textrm{val}(f) = R\), but the problems are of the same complexity, cf. [28].

  2. 2.

    In the full version [8], we in fact give a detailed, slightly different, algorithm for Capacitated Red-Blue Dominating Set and then deduce the result for Capacitated Dominating Set.

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Acknowledgements

We thank Carla Groenland and Hugo Jacob for various discussions, and in particular for suggestions related to the capacitated dominating set problems, and the relations between tree cut-width, tree partition width, and stable tree-partition width.

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Authors and Affiliations

  1. Department of Information and Computing Sciences, Utrecht University, Princetonplein 5, 3584CC, Utrecht, The Netherlands

    Hans L. Bodlaender

  2. Department of Mathematics, Utrecht University, P.O. Box 80010, 3508 TA, Utrecht, The Netherlands

    Gunther Cornelissen & Marieke van der Wegen

Authors
  1. Hans L. Bodlaender
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  2. Gunther Cornelissen
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  3. Marieke van der Wegen
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Correspondence to Hans L. Bodlaender .

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Editors and Affiliations

  1. University of Ioannina, Ioannina, Greece

    Michael A. Bekos

  2. Universität Tübingen, Tübingen, Germany

    Michael Kaufmann

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Bodlaender, H.L., Cornelissen, G., van der Wegen, M. (2022). Problems Hard for Treewidth but Easy for Stable Gonality. In: Bekos, M.A., Kaufmann, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2022. Lecture Notes in Computer Science, vol 13453. Springer, Cham. https://doi.org/10.1007/978-3-031-15914-5_7

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  • DOI: https://doi.org/10.1007/978-3-031-15914-5_7

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