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Algorithmica

, Volume 76, Issue 1, pp 279–296 | Cite as

Parameterized Algorithms for Non-separating Trees and Branchings in Digraphs

  • Jørgen Bang-Jensen
  • Saket Saurabh
  • Sven Simonsen
Article

Abstract

A well known result in graph algorithms, due to Edmonds, states that given a digraph D and a positive integer \(\ell \), we can test whether D contains \(\ell \) arc-disjoint out-branchings in polynomial time. However, if we ask whether there exists an out-branching and an in-branching which are arc-disjoint, then the problem becomes NP-complete. In fact, even deciding whether a digraph D contains an out-branching which is arc-disjoint from some spanning tree in the underlying undirected graph remains NP-complete. In this paper we formulate some natural optimization questions around these problems and initiate its study in the realm of parameterized complexity. More precisely, the problems we study are the following: Arc-Disjoint Branchings and Non-Disconnecting Out-Branching. In Arc-Disjoint Branchings (Non-Disconnecting Out-Branching), a digraph D and a positive integer k are given as input and the goal is to test whether there exist an out-branching and in-branching (respectively, a spanning tree in the underlying undirected graph) that differ on at least k arcs. We obtain the following results for these problems.
  • Non-Disconnecting Out-Branching is fixed parameter tractable (FPT) and admits a linear vertex kernel.

  • Arc-Disjoint Branchings is FPT on strong digraphs.

The algorithm for Non-Disconnecting Out-Branching runs in time \(2^{\mathcal {O}(k)}n^{\mathcal {O}(1)}\) and the approach we use to obtain this algorithms seems useful in designing other moderately exponential time algorithms for edge/arc partitioning problems.

Keywords

Branching Spanning tree Fixed parameter tractable Parameterized complexity Linear vertex kernel Exponential time algorithm Partitioning problem 

Notes

Acknowledgments

We would like to thank the reviewers for their valuable comments and suggestions.

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Jørgen Bang-Jensen
    • 1
  • Saket Saurabh
    • 2
  • Sven Simonsen
    • 1
  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdenseDenmark
  2. 2.The Institute of Mathematical SciencesChennaiIndia

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