A Faster FPT Algorithm for Finding Spanning Trees with Many Leaves

Bonsma, Paul; Brueggemann, Tobias; Woeginger, Gerhard J.

doi:10.1007/978-3-540-45138-9_20

Cited by 35 publications

(38 citation statements)

References 13 publications

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“…Since then, considerable effort has been put in finding faster FPT algorithms for this problem, see e.g. [4,7,3,6,2]. The papers [3,6,2] also establish a strong connection between extremal graph-theoretic results and fast FPT algorithms.…”

Section: ℓ(T )mentioning

confidence: 99%

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Spanning Trees with Many Leaves in Graphs without Diamonds and Blossoms

Bonsma

Zickfeld

Lecture Notes in Computer Science

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It is known that graphs on n vertices with minimum degree at least 3 have spanning trees with at least n/4 + 2 leaves and that this can be improved to (n + 4)/3 for cubic graphs without the diamond K 4 − e as a subgraph. We generalize the second result by proving that every graph with minimum degree at least 3, without diamonds and certain subgraphs called blossoms, has a spanning tree with at least (n+4)/3 leaves, and generalize this further by allowing vertices of lower degree. We show that it is necessary to exclude blossoms in order to obtain a bound of the form n/3 + c.We use the new bound to obtain a simple FPT algorithm, which decides in O(m) + O * (6.75 k ) time whether a graph of size m has a spanning tree with at least k leaves. This improves the best known time complexity for Max Leaf Spanning Tree.

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Section: ℓ(T )mentioning

confidence: 99%

“…[4,7,3,6,2]. The papers [3,6,2] also establish a strong connection between extremal graph-theoretic results and fast FPT algorithms. In [3], the bound of n/4+ 2 from [11] This algorithm is the fastest FPT algorithm for MaxLeaf at the moment, both optimizing the dependency on the input size and the parameter function.…”

Section: ℓ(T )mentioning

confidence: 99%

Spanning Trees with Many Leaves in Graphs without Diamonds and Blossoms

Bonsma

Zickfeld

Lecture Notes in Computer Science

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“…Bonsma, Brueggemann, and Woeginger [5] use an involved result from extremal graph theory by Linial and Sturtevant [19], and Kleitman and West [17] to bound the number of nodes that can possibly be leaves by 4k. A brute force check for each k-subset of these 4k nodes yields a run time bound of O(|V | 3 +9.4815 k k 3 ).…”

Section: Introductionmentioning

confidence: 99%

A New Algorithm for Finding Trees with Many Leaves

2010

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Abstract. We present an algorithm that finds trees with at least k leaves in undirected and directed graphs. These problems are known as Maximum Leaf Spanning Tree for undirected graphs, and, respectively, Directed Maximum Leaf Out-Tree and Directed Maximum Leaf Spanning Out-Tree in the case of directed graphs. The run time of our algorithm is O(poly(|V |) + 4 k k 2 ) on undirected graphs, and O(4 k |V |·|E|) on directed graphs. Currently, the fastest algorithms for these problems have run times of O(poly(n) + 6.75 k poly(k)) and 2 O(k log k) poly(n), respectively.

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“…Unlike its undirected counterpart which has attracted a lot of attention in all algorithmic paradigms like approximation algorithms [24,31,33], parameterized algorithms [11,19,21], exact exponential time algorithms [20] and also combinatorial studies [16,25,27,30], the Directed Maximum Leaf Out-Branching problem has largely been neglected until recently. The only paper we are aware of is the very recent paper [18] that describes an O( √ opt)-approximation algorithms for DMLOB.…”

Section: Introductionmentioning

confidence: 99%

Spanning Directed Trees with Many Leaves

Alon¹,

Fomin²,

Gutin³

et al. 2009

SIAM J. Discrete Math.

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Abstract. The Directed Maximum Leaf Out-Branching problem is to find an out-branching (i.e. a rooted oriented spanning tree) in a given digraph with the maximum number of leaves. In this paper, we obtain two combinatorial results on the number of leaves in out-branchings. We show that -every strongly connected n-vertex digraph D with minimum indegree at least 3 has an out-branching with at least (n/4) 1/3 − 1 leaves; -if a strongly connected digraph D does not contain an out-branching with k leaves, then the pathwidth of its underlying graph UG(D) is O(k log k). Moreover, if the digraph is acyclic, the pathwidth is at most 4k. The last result implies that it can be decided in time 2whether a strongly connected digraph on n vertices has an out-branching with at least k leaves. On acyclic digraphs the running time of our algorithm is 2 O(k log k) · n O(1) .

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A Faster FPT Algorithm for Finding Spanning Trees with Many Leaves

Cited by 35 publications

References 13 publications

Spanning Trees with Many Leaves in Graphs without Diamonds and Blossoms

Spanning Trees with Many Leaves in Graphs without Diamonds and Blossoms

A New Algorithm for Finding Trees with Many Leaves

Spanning Directed Trees with Many Leaves

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