On the Complexity of Finding Internally Vertex-Disjoint Long Directed Paths

  • Júlio Araújo
  • Victor A. Campos
  • Ana Karolinna Maia
  • Ignasi SauEmail author
  • Ana Silva
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10807)


For two positive integers k and \(\ell \), a \((k \times \ell )\) -spindle is the union of k pairwise internally vertex-disjoint directed paths with \(\ell \) arcs each between two vertices u and v. We are interested in the (parameterized) complexity of several problems consisting in deciding whether a given digraph contains a subdivision of a spindle, which generalize both the Maximum Flow and Longest Path problems. We obtain the following complexity dichotomy: for a fixed \(\ell \ge 1\), finding the largest k such that an input digraph G contains a subdivision of a \((k \times \ell )\)-spindle is polynomial-time solvable if \(\ell \le 3\), and NP-hard otherwise. We place special emphasis on finding spindles with exactly two paths and present FPT algorithms that are asymptotically optimal under the ETH. These algorithms are based on the technique of representative families in matroids, and use also color-coding as a subroutine. Finally, we study the case where the input graph is acyclic, and present several algorithmic and hardness results.


Digraph subdivision Spindle Parameterized complexity FPT algorithm Representative family Complexity dichotomy 


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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Júlio Araújo
    • 1
  • Victor A. Campos
    • 2
  • Ana Karolinna Maia
    • 2
  • Ignasi Sau
    • 1
    • 3
    Email author
  • Ana Silva
    • 1
  1. 1.ParGO Research Group, Departamento de MatemáticaUniversidade Federal do CearáFortalezaBrazil
  2. 2.ParGO Research Group, Departamento de ComputaçãoUniversidade Federal do CearáFortalezaBrazil
  3. 3.CNRS, Université de Montpellier, LIRMMMontpellierFrance

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