Tight Bounds and a Fast FPT Algorithm for Directed Max-Leaf Spanning Tree

  • Authors:

  • Paul Bonsma;Frederic Dorn

  • Affiliations:

  • Institut für Mathematik, Technische Universität Berlin, Berlin, Germany 10623;Institut für Informatik, Humboldt-Universität zu Berlin, Berlin, Germany 10099

  • Venue:
  • ESA '08 Proceedings of the 16th annual European symposium on Algorithms
  • Year:
  • 2008

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Abstract

An out-tree Tof a directed graph Dis a rooted tree subgraph with all arcs directed outwards from the root. An out-branching is a spanning out-tree. By 茂戮驴(D) and 茂戮驴s(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 茂戮驴s(D) 茂戮驴 kand whether 茂戮驴(D) 茂戮驴 kfor a digraph Don nvertices, both with time complexity 2O(klogk)·nO(1). This proves the problem for out-branchings to be in FPT, and improves on the previous complexity of $2^{O(k\log^2 k)} \cdot n^{O(1)}$ for out-trees. To obtain the algorithm for out-branchings, we prove that when all arcs of Dare part of at least one out-branching, 茂戮驴s(D) 茂戮驴 茂戮驴(D)/3. The second bound we prove states that for strongly connected digraphs Dwith minimum in-degree 3, $\ell_s(D)\geq \Theta(\sqrt{n})$, where previously $\ell_s(D)\geq \Theta(\sqrt[3]{n})$ 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.