An Exact Algorithm for the Maximum Leaf Spanning Tree Problem

Abstract

Abstract. Given an undirected graph with n nodes, the Maximum Leaf Spanning Tree problem is to find a spanning tree with as many leaves as possible. When parameterized in the number of leaves k, this problem can be solved in time O(4 k poly(n)) using a simple branching algorithm introduced by a subset of the authors [12]. Daligault, Gutin, Kim, and Yeo [6] improved the branching and obtained a running time of O(3.72 k poly(n)). In this paper, we study the problem from an exponential time viewpoint, where it is equivalent to the Connected Dominating Set problem. Here, Fomin, Grandoni, and Kratsch showed how to break the Ω(2 n ) barrier and proposed an O(1.9407 n )-time algorithm [10]. In light of some useful properties of [12] and [6], we present a branching algorithm whose running time of O(1.8966 n ) has been analyzed using the Measure-and-Conquer technique. Finally we provide a lower bound of Ω(1.4422 n ) for the worst case running time of our algorithm.

An Exact Exponential Time Algorithm for Power Dominating Set

“…Further results and general explanations on the Measure&Conquer paradigm can be found in [6]. Under the alternative name MAXIMUM LEAF SPANNING TREE, this upper bound was improved to O * (1.8966 n ) by Fernau et al [5]; a further small improvement was reported in [17]. Brueni and Heath [2] showed that there exists a power dominating set of size at most n/3 for any graph with at least three vertices.…”

An exact algorithm for connected red–blue dominating set

“…The current best-known algorithm was presented recently in [7] where the authors solve cds by reduction to the Maximum-Leaf Spanning Tree problem, in O * (1.8966 n ). The following reduction to crbds yields a better worst-case run-time.…”

Matheuristics for Variants of the Dominating Set Problem

“…The MCDS problem is equivalent to finding a spanning tree with the maximum number of leaves in the graph (Fernau et al, 2011). The Maximum Leaf Spanning Tree (MLST) problem aims is to find a spanning tree with as many leaves as possible.…”

“…Some exact approaches have been developed for the MCDS problem, such as those of Fujie (2003), Lucena et al (2010), Fernau et al (2011), and Gendron et al (2014. In more details, Lucena et al (2010) introduced two integer programming formulations: the first one is based on a Steiner reformulation of the problem, and the second considers the problem in a directed graph, seeking later a spanning arborescence with as many leaves as possible.…”