License: Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.ESA.2023.65
URN: urn:nbn:de:0030-drops-187184
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2023/18718/
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Jacob, Ashwin ; Włodarczyk, Michał ; Zehavi, Meirav

Finding Long Directed Cycles Is Hard Even When DFVS Is Small or Girth Is Large

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LIPIcs-ESA-2023-65.pdf (7 MB)

Abstract

We study the parameterized complexity of two classic problems on directed graphs: Hamiltonian Cycle and its generalization Longest Cycle. Since 2008, it is known that Hamiltonian Cycle is W[1]-hard when parameterized by directed treewidth [Lampis et al., ISSAC'08]. By now, the question of whether it is FPT parameterized by the directed feedback vertex set (DFVS) number has become a longstanding open problem. In particular, the DFVS number is the largest natural directed width measure studied in the literature. In this paper, we provide a negative answer to the question, showing that even for the DFVS number, the problem remains W[1]-hard. As a consequence, we also obtain that Longest Cycle is W[1]-hard on directed graphs when parameterized multiplicatively above girth, in contrast to the undirected case. This resolves an open question posed by Fomin et al. [ACM ToCT'21] and Gutin and Mnich [arXiv:2207.12278]. Our hardness results apply to the path versions of the problems as well. On the positive side, we show that Longest Path parameterized multiplicatively above girth belongs to the class XP.

BibTeX - Entry

@InProceedings{jacob_et_al:LIPIcs.ESA.2023.65,
  author =	{Jacob, Ashwin and W{\l}odarczyk, Micha{\l} and Zehavi, Meirav},
  title =	{{Finding Long Directed Cycles Is Hard Even When DFVS Is Small or Girth Is Large}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{65:1--65:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. and Herman, Grzegorz},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/opus/volltexte/2023/18718},
  URN =		{urn:nbn:de:0030-drops-187184},
  doi =		{10.4230/LIPIcs.ESA.2023.65},
  annote =	{Keywords: Hamiltonian cycle, longest path, directed feedback vertex set, directed graphs, parameterized complexity}
}

Keywords: Hamiltonian cycle, longest path, directed feedback vertex set, directed graphs, parameterized complexity
Collection: 31st Annual European Symposium on Algorithms (ESA 2023)
Issue Date: 2023
Date of publication: 30.08.2023

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