An exact algorithm for the Maximum Leaf Spanning Tree problem
Résumé
Given an undirected graph G with n nodes, the Maximum Leaf Spanning Tree problem asks to find a spanning tree of G 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 [Kneis, Langer, Rossmanith, A new algorithm for finding trees with many leaves, ISAAC'2008, LNCS 5369, pages 270-281]. Daligault, Gutin, Kim, and Yeo [Daligault, Gutin, Kim, Yeo, FPT Algorithms and Kernels for the Directed k-Leaf Problem, CoRR abs/0810.4946, 2008; J. Comput. System Sci. (2009), doi:10.1016/j.jcss.2009.06.005] improved this branching algorithm and obtained a running time of $O(3.72^k poly(n))$. In this paper, we study the problem from an exact exponential time point of view, where it is equivalent to the Connected Dominating Set problem. For this problem Fomin, Grandoni, and Kratsch showed how to break the $\Omega(2^n)$ barrier and proposed an $O(1.9407^n)$ time algorithm [Fomin, Grandoni, Kratsch, Solving connected dominating set faster than $2^n$, Algorithmica, 52(2):153--166, 2008]. Based on some properties of [Daligault, Gutin, Kim, Yeo, FPT Algorithms and Kernels for the Directed k-Leaf Problem, CoRR abs/0810.4946, 2008; J. Comput. System Sci. (2009), doi:10.1016/j.jcss.2009.06.005] and [Kneis, Langer, Rossmanith, A new algorithm for finding trees with many leaves, ISAAC'2008, LNCS 5369, pages 270-281], we establish 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 $\Omega(1.4422^n)$ for the worst case running time of our algorithm.