An approximation algorithm for the maximum leaf spanning arborescence problem

Drescher, Matthew; Vetta, Adrian

doi:10.1145/1798596.1798599

Cited by 21 publications

(16 citation statements)

References 27 publications

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“…The algorithms developed for such problems yield a solution of value g(k) for a problem parameterized by k, where k is the solution size (see, e.g., [20,16,18]). …”

Section: Related Workmentioning

confidence: 99%

Parameterized Approximation via Fidelity Preserving Transformations

Fellows

Kulik

Rosamond

et al. 2012

Automata, Languages, and Programming

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Abstract.We motivate and describe a new parameterized approximation paradigm which studies the interaction between performance ratio and running time for any parameterization of a given optimization problem. As a key tool, we introduce the concept of α-shrinking transformation, for α ≥ 1. Applying such transformation to a parameterized problem instance decreases the parameter value, while preserving approximation ratio of α (or α-fidelity). For example, it is well-known that Vertex Cover cannot be approximated within any constant factor better than 2 [22] (under usual assumptions). Our parameterized α-approximation algorithm for k-Vertex Cover, parameterized by the solution size, has a running time of 1.273(2−α)k , where the running time of the best FPT algorithm is 1.273 k [10]. Our algorithms define a continuous tradeoff between running times and approximation ratios, allowing practitioners to appropriately allocate computational resources. Moving even beyond the performance ratio, we call for a new type of approximative kernelization race. Our α-shrinking transformations can be used to obtain kernels which are smaller than the best known for a given problem. For the Vertex Cover problem we obtain a kernel size of 2(2 − α)k. The smaller "α-fidelity" kernels allow us to solve exactly problem instances more efficiently, while obtaining an approximate solution for the original instance. We show that such transformations exist for several fundamental problems, including Vertex Cover, d-Hitting Set, Connected Vertex Cover and Steiner Tree. We note that most of our algorithms are easy to implement and are therefore practical in use.

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“…The algorithms developed for such problems yield a solution of value g(k) for a problem parameterized by k, where k is the solution size (see, e.g., [20,16,18]). …”

Section: Related Workmentioning

confidence: 99%

Parameterized Approximation via Fidelity Preserving Transformations

Fellows

Kulik

Rosamond

et al. 2012

Automata, Languages, and Programming

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“…The only known approximation algorithm for Directed Max Leaf is due to Drescher and Vetta [12] and its approximation…”

Section: Introductionmentioning

confidence: 99%

FPT algorithms and kernels for the Directed k-Leaf problem

Daligault

Gutin

Kim

et al. 2010

Journal of Computer and System Sciences

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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.

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“…The important contribution here is Lemma 5.1, which gives a strong bound for the number of leaves that can be obtained for out-branchings, and implicitly a corresponding polynomial-time algorithm. We expect that (a generalization of) this lemma can be used to answer other important algorithmic questions on out-branchings [Gutin 2009], such as: is it possible to improve the √ OPTapproximation ratio for MAX-LEAF OUT-BRANCHING from Drescher and Vetta [2010]? Does there exist a ck-kernelization algorithm for rooted-k-LEAF OUT-BRANCHING for some constant c (see Fernau et al [2009])?…”

Section: Discussionmentioning

confidence: 99%

“…Whereas for the undirected problem, MAX-LEAF SPANNING TREE, a 2-approximation is known [Solis-Oba 1998], the best known approximation result for MAX-LEAF OUT-BRANCHING is a very recent algorithm with ratio O( √ OPT) [Drescher and Vetta 2010]. In the algorithmic part of this work, we are interested in fixed parameter tractable (FPT) algorithms for the decision problems.…”

Section: Introductionmentioning

confidence: 99%

Tight bounds and a fast FPT algorithm for directed Max-Leaf Spanning Tree

Bonsma

Dorn

2011

ACM Trans. Algorithms

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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 (D) and 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 s (D) ≥ k and whether (D) ≥ k for a digraph D on n vertices, both with time complexity 2 O(k log k) · n O(1) . This answers an open question whether the problem for out-branchings is in FPT, and improves on the previous complexity of 2 O(k log 2 k) · n O(1) in the case of out-trees.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, s (D) ≥ (D)/3. The second bound we prove in this article states that for strongly connected digraphs D with minimum in-degree 3, s (D) ≥ ( √ n), where previously s (D) ≥ ( 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. ACM Reference Format:Bonsma, P. and Dorn, F. 2011. Tight bounds and a fast FPT algorithm for directed max-leaf spanning tree ACM Trans.

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An approximation algorithm for the maximum leaf spanning arborescence problem

Cited by 21 publications

References 27 publications

Parameterized Approximation via Fidelity Preserving Transformations

Parameterized Approximation via Fidelity Preserving Transformations

FPT algorithms and kernels for the Directed k-Leaf problem

Tight bounds and a fast FPT algorithm for directed Max-Leaf Spanning Tree

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