A New Algorithm for Finding Trees with Many Leaves

Kneis, Joachim; Langer, Alexander; Rossmanith, Peter

doi:10.1007/s00453-010-9454-5

Cited by 22 publications

(45 citation statements)

References 32 publications

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“…This approach carries over to the weighted setting by building weightaware reduction rules that eliminate the diamonds and blossoms. It is an open question whether the O(4 k n O(1) ) FPT algorithm of Kneis et al [10] can be adapted to solve the weighted problem.…”

Section: Resultsmentioning

confidence: 99%

Kernelization for Maximum Leaf Spanning Tree with Positive Vertex Weights

Jansen

2010

Lecture Notes in Computer Science

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Abstract. In this paper we consider a natural generalization of the well-known Max Leaf Spanning Tree problem. In the generalized Weighted Max Leaf problem we get as input an undirected connected graph G, a rational number k not smaller than 1 and a weight function w : V → Q ≥1 on the vertices, and are asked whether a spanning tree T for G exists such that the combined weight of the leaves of T is at least k. We show that it is possible to transform an instance G, w, k of Weighted Max Leaf in linear time into an equivalent instance G , w , k such that |V (G )| ≤ 5.5k and k ≤ k. In the context of fixed parameter complexity this means that Weighted Max Leaf admits a kernel with 5.5k vertices. The analysis of the kernel size is based on a new extremal result which shows that every graph G = (V, E) that excludes some simple substructures always contains a spanning tree with at least |V |/5.5 leaves. 1

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Section: Resultsmentioning

confidence: 99%

Kernelization for Maximum Leaf Spanning Tree with Positive Vertex Weights

Jansen

2010

Lecture Notes in Computer Science

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“…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.…”

Section: Introductionmentioning

confidence: 99%

“…In the next sections we solve the MLST problem in time O (1.8966 n ), improving the result of [10]. The algorithm presented here is based on the parameterized one [12], which basically repeatedly branches on leaves of a subtree of the graph in order to decide whether it can remain a leaf or must become an internal node. 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.…”

Section: Introductionmentioning

confidence: 99%

“…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.…”

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confidence: 99%

“…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)).…”

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confidence: 99%

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An Exact Algorithm for the Maximum Leaf Spanning Tree Problem

Fernau

Kneis

Kratsch³

et al. 2009

Parameterized and Exact Computation

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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.

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Finding k-Secluded Trees Faster

Donkers

Jansen

Kroon

2022

Graph-Theoretic Concepts in Computer Science

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We revisit the k-Secluded Tree problem. Given a vertex-weighted undirected graph G, its objective is to find a maximum-weight induced subtree T whose open neighborhood has size at most k. We present a fixed-parameter tractable algorithm that solves the problem in time $$2^{\mathcal {O} (k \log k)}\cdot n^{\mathcal {O} (1)}$$, improving on a double-exponential running time from earlier work by Golovach, Heggernes, Lima, and Montealegre. Starting from a single vertex, our algorithm grows a k-secluded tree by branching on vertices in the open neighborhood of the current tree T. To bound the branching depth, we prove a structural result that can be used to identify a vertex that belongs to the neighborhood of any k-secluded supertree $$T' \supseteq T$$ once the open neighborhood of T becomes sufficiently large. We extend the algorithm to enumerate compact descriptions of all maximum-weight k-secluded trees, which allows us to count the number of such trees containing a specified vertex in the same running time.

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A New Algorithm for Finding Trees with Many Leaves

Cited by 22 publications

References 32 publications

Kernelization for Maximum Leaf Spanning Tree with Positive Vertex Weights

Kernelization for Maximum Leaf Spanning Tree with Positive Vertex Weights

An Exact Algorithm for the Maximum Leaf Spanning Tree Problem

Finding k-Secluded Trees Faster

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