On the directed Full Degree Spanning Tree problem

  • Authors:

  • Daniel Lokshtanov;Venkatesh Raman;Saket Saurabh;Somnath Sikdar

  • Affiliations:

  • The University of Bergen, Norway;The Institute of Mathematical Sciences, India;The University of Bergen, Norway;The Institute of Mathematical Sciences, India

  • Venue:
  • Discrete Optimization
  • Year:
  • 2011

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Abstract

We study the parameterized complexity of a directed analog of the Full Degree Spanning Tree problem where, given a digraph D and a nonnegative integer k, the goal is to construct a spanning out-tree T of D such that at least k vertices in T have the same out-degree as in D. We show that this problem is W[1]-hard even on the class of directed acyclic graphs. In the dual version, called Reduced Degree Spanning Tree, one is required to construct a spanning out-tree T such that at most k vertices in T have out-degrees that are different from that in D. We show that this problem is fixed-parameter tractable and that it admits a problem kernel with at most 8k vertices on strongly connected digraphs and O(k^2) vertices on general digraphs. We also give an algorithm for this problem on general digraphs with running time O(5.942^k@?n^O^(^1^)), where n is the number of vertices in the input digraph.