Tight bounds and a fast FPT algorithm for directed Max-Leaf Spanning Tree

  • Authors:

  • Paul Bonsma;Frederic Dorn

  • Affiliations:

  • Technische Universität Berlin, Germany;Humboldt-Universität zu Berlin, Germany

  • Venue:
  • ACM Transactions on Algorithms (TALG)
  • Year:
  • 2011

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Abstract

An out-tree T of a directed graph D is a rooted tree subgraph with all arcs directed outwards from the root. An out-branching is a spanning out-tree. By ℓ(D) and ℓs(D), we denote the maximum number of leaves over all out-trees and out-branchings of D, respectively. We give fixed parameter tractable algorithms for deciding whether ℓs(D) ≥ k and whether ℓ(D) ≥ k for a digraph D on n vertices, both with time complexity 2O(k log k) · nO(1). This answers an open question whether the problem for out-branchings is in FPT, and improves on the previous complexity of 2O(klog 2 k) · nO(1) in the case of out-trees. To obtain the complexity bound in the case of out-branchings, we prove that when all arcs of D are part of at least one out-branching, ℓs (D) ≥ ℓ(D)/3. The second bound we prove in this article states that for strongly connected digraphs D with minimum in-degree 3, ℓs(D) ≥ Θ(&sqrt;n), where previously ℓs(D) ≥ Θ(3&sqrt;n) was the best known bound. This bound is tight, and also holds for the larger class of digraphs with minimum in-degree 3 in which every arc is part of at least one out-branching.