Better Algorithms and Bounds for Directed Maximum Leaf Problems

TL;DR

This paper advances algorithms and bounds for the directed maximum leaf out-branching problem, providing new combinatorial bounds, structural insights, and fixed-parameter tractable algorithms for finding out-branchings with many leaves.

Contribution

It introduces improved bounds and algorithms for the directed maximum leaf out-branching problem, including new combinatorial bounds and fixed-parameter algorithms.

Findings

Strongly connected digraphs with minimum in-degree 3 have large out-branchings.

Pathwidth of underlying graph is bounded by O(k log k) if no out-branching with k leaves exists.

Deciding the existence of an out-branching with at least k leaves is fixed-parameter tractable.

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.