On Finding Directed Trees with Many Leaves

Abstract

International audienceThe Rooted Maximum Leaf problem consists in finding a spanning directed tree rooted at some prescribed vertex of a digraph with the maximum number of leaves. Its parameterized version asks if there exists such a tree with at least $k$ leaves. We use the notion of $s-t$ numbering studied in \cite{stnum, stnumdir,LLWemb} to exhibit combinatorial bounds on the existence of spanning directed trees with many leaves. These combinatorial bounds allow us to produce a constant factor approximation algorithm for finding directed trees with many leaves, whereas the best known approximation algorithm has a $\sqrt{OPT}$-factor \cite{DrescherMaxLeaf}. We also show that Rooted Maximum Leaf admits an edge-quadratic kernel, improving over the vertex-cubic kernel given by Fernau et al \cite{FernauMaxLeaf}

A Linear-Time Kernelization for the Rooted k-Leaf Outbranching Problem

“…In contrast, we use the separator rule only if we already know the separator S. Moreover, a single application of the dominator rule, which is used by Daligault and Thomassé [6], can add a quadratic number of edges and, hence, by itself incurs a quadratic running time. Informally speaking, the new shortcut rule allows us to do the modifications of the dominator rule one by one.…”

“…In the same year, Daligault and Thomassé [6] described an algorithm to compute a kernel of size O(k 2 ) for the Rooted k-Leaf Outbranching Problem. An efficient implementation of the algorithm runs in quadratic time.…”

“…An edge (u, v) is useless if v is a dominator for u, and useful otherwise. A list L of The correctness of Rules 1-5 follows from [4,6]. Consider the situation described in Rule 6.…”

“…Our algorithm is based on one new and several well-known reduction rules [4,6], which we apply in a certain order. Assume that we are given an instance of the parameterized Rooted kLeaf Outbranching Problem, i.e., a digraph G = (V, E), r ∈ V, and k ≥ 2.…”

“…We present a linear-time algorithm that, given a general digraph G and a parameter k ∈ IN , computes a kernel of (G, k) with O(k 6 ) vertices and O(k 7 ) edges for the Rooted k-Leaf Outbranching Problem. As usual, an outtree T is a rooted tree in which all edges are directed away from a vertex r called the root.…”