Tight Bounds and Faster Algorithms for Directed Max-Leaf Problems

Paul Bonsma and Frederic Dorn

submitted

not available

spanning tree, max leaf, directed graph, fixed parameter tractable

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 l_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 l_s(D)>=k and whether l(D)>=k for a digraph D on n vertices, both with time complexity 2^{O(k log k)} n^O(1). This improves on previous algorithms with complexity 2^{O(k^3 log k)} n^O(1) and 2^{O(k log^2 k)} n^O(1), respectively.

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, l_s(D)>=l(D)/3. The second bound we prove in this paper states that for strongly connected digraphs D with minimum in-degree 3, l_s(D)>= Θ (n^1/2), where previously l_s(D)>= Θ (n^1/3) 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.

Created on May 07 2008

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