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Algorithmica

, Volume 76, Issue 2, pp 320–343 | Cite as

Algorithms and Kernels for Feedback Set Problems in Generalizations of Tournaments

  • Jørgen Bang-Jensen
  • Alessandro Maddaloni
  • Saket Saurabh
Article

Abstract

In the Directed Feedback Arc (Vertex) Set problem, we are given a digraph D with vertex set V(D) and arcs set A(D) and a positive integer k, and the question is whether there is a subset X of arcs (vertices) of size at most k such that the digraph obtained after deleting X from D is an acyclic digraph. In this paper we study these two problems in the realm of parametrized and kernelization complexity. More precisely, for these problems we give polynomial time algorithms, known as kernelization algorithms, on several digraph classes that given an instance (Dk) of the problem returns an equivalent instance \((D',k')\) such that the size of \(D'\) and \(k'\) is at most \(k^{O(1)}\). We extend previous results for Directed Feedback Arc (Vertex) Set on tournaments to much larger and well studied classes of digraphs. Specifically we obtain polynomial kernels for k-FVS on digraphs with bounded independence number, locally semicomplete digraphs and some totally \(\Phi \)-decomposable digraphs, including quasi-transitive digraphs. We also obtain polynomial kernels for k-FAS on some totally \(\Phi \)-decomposable digraphs, including quasi-transitive digraphs. Finally, we design a subexponential algorithm for k-FAS running in time \(2^{O(\sqrt{k} (\log k)^c)}n^d\) for constants cd. on locally semicomplete digraphs.

Keywords

Parameterized complexity Kernels Feedback vertex set Feedback arc set Decomposable digraph Bounded independence number Locally semicomplete digraph Quasi-transitive digraph 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Jørgen Bang-Jensen
    • 1
  • Alessandro Maddaloni
    • 2
  • Saket Saurabh
    • 3
  1. 1.Department of Mathematics and Computer ScienceUniversity of Southern DenmarkOdenseDenmark
  2. 2.TeCIP Institute, Scuola Superiore Sant’AnnaPisaItaly
  3. 3.The Institute of Mathematical SciencesChennaiIndia

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