Tight Bounds and Faster Algorithms for Directed Max-Leaf Problems

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 $l_s(D)\geq k$andwhether$l(D)\geq k$foradigraph$D$on$n$ vertices, both with time complexity $2^{O(k\log k)}\cdot n^{O(1)}$. This improves on previous algorithms with complexity $2^{O(k^3\log k)}\cdot n^{O(1)}$ and $2^{O(k\log^2 k)} \cdot n^{O(1)}$, 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, $l_s(D)\geq l(D)/3$. The second bound we prove in this paper states that for strongly connected digraphs $D$ with minimum in-degree 3, $l_s(D)\ge \mathrm{\Theta }(\sqrt{n})$, where previously $l_s(D)\ge \mathrm{\Theta }(\sqrt[3]{n})$ 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.