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arXiv · 0707.1095

Better Algorithms and Bounds for Directed Maximum Leaf Problems

Noga Alon·Fedor V. Fomin·Gregory Gutin·Michael Krivelevich·Saket Saurabh
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Abstract

The {\sc Directed Maximum Leaf Out-Branching} problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we improve known parameterized algorithms and combinatorial bounds on the number of leaves in out-branchings. We show that \begin{itemize} \item every strongly connected digraph $D$ of order $n$ with minimum in-degree at least 3 has an out-branching with at least $(n/4)^{1/3}-1$ leaves; \item if a strongly connected digraph $D$ does not contain an out-branching with $k$ leaves, then the pathwidth of its underlying graph is $O(k\log k)$; \item it can be decided in time $2^{O(k\log^2 k)}\cdot n^{O(1)}$ whether a strongly connected digraph on $n$ vertices has an out-branching with at least $k$ leaves. \end{itemize} All improvements use properties of extremal structures obtained after applying local search and of some out-branching decompositions.

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BibTeXRIS

Noga Alon, Fedor V. Fomin, Gregory Gutin, Michael Krivelevich, Saket Saurabh. 2007-07-07. Better Algorithms and Bounds for Directed Maximum Leaf Problems. https://arxiv.org/abs/0707.1095

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