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

Authors Ashwin Jacob , Michał Włodarczyk , Meirav Zehavi

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Ashwin Jacob
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel
Michał Włodarczyk
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel
Meirav Zehavi
  • Ben-Gurion University of the Negev, Beer-Sheva, Israel

Funding

European Research Council (ERC) grant titled PARAPATH.

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Ashwin Jacob, Michał Włodarczyk, and Meirav Zehavi. Finding Long Directed Cycles Is Hard Even When DFVS Is Small or Girth Is Large. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 65:1-65:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.65

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.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Hamiltonian cycle
  • longest path
  • directed feedback vertex set
  • directed graphs
  • parameterized complexity

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