FPT algorithms and kernels for the Directed k-Leaf problem

Abstract

A subgraph T of a digraph D is an out-branching if T is an oriented spanning tree with only one vertex of in-degree zero (called the root). The vertices of T of out-degree zero are leaves. In the Directed Max Leaf problem, we wish to find the maximum number of leaves in an out-branching of a given digraph D (or, to report that D has no out-branching). In the Directed k-Leaf problem, we are given a digraph D and an integral parameter k, and we are to decide whether D has an out-branching with at least k leaves. Recently, Kneis et al. (2008) obtained an algorithm for Directed k-Leaf of running time 4 k · n O (1) . We describe a new algorithm for Directed k-Leaf of running time 3.72 k · n O (1) . This algorithms leads to an O (1.9973 n )-time algorithm for solving Directed Max Leaf on a digraph of order n. The latter algorithm is the first algorithm of running time O (γ n ) for Directed Max Leaf, where γ < 2. In the Rooted Directed k-Leaf problem, apart from D and k, we are given a vertex r of D and we are to decide whether D has an out-branching rooted at r with at least k leaves. Very recently, Fernau et al. (2008) found an O (k 3 )-size kernel for Rooted Directed k-Leaf. In this paper, we obtain an O (k) kernel for Rooted Directed k-Leaf restricted to acyclic digraphs.

An Exact Algorithm for the Maximum Leaf Spanning Tree Problem

“…It enables us to turn some nodes into additional floating leaves in some special cases. A similar technique has already been used in [6].…”

“…There is however a long research history for this problem in the field of parameterized complexity, see [1,7,9,3,8,2,4]. The currently fastest published algorithm is due to Kneis, Langer, and Rossmanith [12] with a runtime bounded by O * (4 k ), which has been further improved to O * (3.72 k ) by Daligault, Gutin, Kim, and Yeo in an yet to appear article (a preliminary version can be found in [6]), whose improvements are also used in our exact algorithm.…”

“…If we analyze the running time as a function of n, we find that branching on nodes of small degree (with two possible successors) becomes the worst case resulting in a bad running time. This resembles the worst case of the parameterized algorithm, and the changes in [6] are based on improving exactly this case. We use a similar approach for our exact algorithm.…”

On the Directed Degree-Preserving Spanning Tree Problem

“…The Full Degree Spanning Tree problem is one of the many variants of the generic Constrained Spanning Tree problem, where one is required to find a spanning tree of a given (di)graph subject to certain constraints. This class of problems has been studied intensely of late [1,7,8,10,11,15,18].…”

“…In the directed variant of this problem, one has to decide whether an input digraph D has an out-branching with at least k leaves. This problem admits a kernel with O(k 3 ) vertices, provided the root of the out-branching is given as part of the input [11], and has an algorithm with run-time O(3.72 k · |V (D)| O(1) ) [8]. Another such problem is Max Internal Spanning Tree, where the objective is to find a spanning tree (or an out-branching, in case of digraphs) with at least k internal vertices.…”