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11
A new algorithm for finding trees with many leaves
, 2001
"... 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 OutTree and Directed Maximum Leaf Spanning OutTree in the case of directed grap ..."
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Cited by 12 (1 self)
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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 OutTree and Directed Maximum Leaf Spanning OutTree 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.
Minimum Leaf OutBranching Problems
 Lect. Notes Comput. Sci. 5034 (2008), 235–246 (Proc. AAIM’08
, 2008
"... Abstract. Given a digraph D, the Minimum Leaf OutBranching problem (MinLOB) is the problem of finding in D an outbranching with the minimum possible number of leaves, i.e., vertices of outdegree 0. We describe three parameterizations of MinLOB and prove that two of them are NPcomplete for every ..."
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Cited by 9 (2 self)
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Abstract. Given a digraph D, the Minimum Leaf OutBranching problem (MinLOB) is the problem of finding in D an outbranching with the minimum possible number of leaves, i.e., vertices of outdegree 0. We describe three parameterizations of MinLOB and prove that two of them are NPcomplete for every value of the parameter, but the third one is fixedparameter tractable (FPT). The FPT parametrization is as follows: given a digraph D of order n and a positive integral parameter k, check whether D contains an outbranching with at most n − k leaves (and find such an outbranching if it exists). We find a problem kernel of order O(k · 16 k) and construct an algorithm of running time O(2 O(k log k) + n 2 log n), which is an ‘additive ’ FPT algorithm. 1
KERNEL(S) FOR PROBLEMS WITH NO KERNEL: ON OUTTREES WITH MANY LEAVES (EXTENDED ABSTRACT)
 STACS 2009
, 2009
"... The kLeaf OutBranching problem is to find an outbranching, that is a rooted oriented spanning tree, with at least k leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the kLea ..."
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Cited by 9 (4 self)
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The kLeaf OutBranching problem is to find an outbranching, that is a rooted oriented spanning tree, with at least k leaves in a given digraph. The problem has recently received much attention from the viewpoint of parameterized algorithms. Here, we take a kernelization based approach to the kLeafOutBranching problem. We give the first polynomial kernel for Rooted kLeafOutBranching, a variant of kLeafOutBranching where the root of the tree searched for is also a part of the input. Our kernel has cubic size and is obtained using extremal combinatorics. For the kLeafOutBranching problem, we show that no polynomial kernel is possible unless the polynomial hierarchy collapses to third level by applying a recent breakthrough result by Bodlaender et al. (ICALP 2008) in a nontrivial fashion. However, our positive results for Rooted kLeafOutBranching immediately imply that the seemingly intractable kLeafOutBranching problem admits a data reduction to n independent O(k³) kernels. These two results, tractability and intractability side by side, are the first ones separating manytoone kernelization from Turing kernelization. This answers affirmatively an open problem regarding “cheat kernelization” raised by Mike Fellows and Jiong Guo independently.
FPT Algorithms and Kernels for the Directed kLeaf Problem
, 2008
"... A subgraph T of a digraph D is an outbranching if T is an oriented spanning tree with only one vertex of indegree zero (called the root). The vertices of T of outdegree zero are leaves. In the Directed kLeaf Problem, we are given a digraph D and an integral parameter k, and we are to decide whet ..."
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Cited by 7 (0 self)
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A subgraph T of a digraph D is an outbranching if T is an oriented spanning tree with only one vertex of indegree zero (called the root). The vertices of T of outdegree zero are leaves. In the Directed kLeaf Problem, we are given a digraph D and an integral parameter k, and we are to decide whether D has an outbranching with at least k leaves. Recently, Kneis et al. (2008) obtained an algorithm for the problem of running time 4 k · n O(1). We describe a new algorithm for the problem of running time 3.72 k · n O(1). In Rooted Directed kLeaf Problem, apart from D and k, we are given a vertex r of D and we are to decide whether D has an outbranching rooted at r with at least k leaves. Very recently, Fernau et al. (2008) found an O(k 3)size kernel for Rooted Directed kLeaf. In this paper, we obtain an O(k) kernel for Rooted Directed kLeaf restricted to acyclic digraphs. 1
Tight bounds and faster algorithms for Directed MaxLeaf
"... An outtree T of a directed graph D is a rooted tree subgraph with all arcs directed outwards from the root. An outbranching is a spanning outtree. By ℓ(D) and ℓs(D) we denote the maximum number of leaves over all outtrees and outbranchings of D, respectively. We give fixed parameter tractable a ..."
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Cited by 6 (0 self)
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An outtree T of a directed graph D is a rooted tree subgraph with all arcs directed outwards from the root. An outbranching is a spanning outtree. By ℓ(D) and ℓs(D) we denote the maximum number of leaves over all outtrees and outbranchings of D, respectively. We give fixed parameter tractable algorithms for deciding whether ℓs(D) ≥ k and whether ℓ(D) ≥ k for a digraph D on n vertices, both with time complexity 2 O(k log k) · n O(1). This improves on previous algorithms with complexity 2 O(k3 log k) · n
Spanning directed trees with many leaves
 SIAM J. Discrete Math
"... Abstract. The Directed Maximum Leaf OutBranching problem is to find an outbranching (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 outbranchings. We show that – every strong ..."
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Cited by 6 (4 self)
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Abstract. The Directed Maximum Leaf OutBranching problem is to find an outbranching (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 outbranchings. We show that – every strongly connected nvertex digraph D with minimum indegree at least 3 has an outbranching with at least (n/4) 1/3 − 1 leaves; – if a strongly connected digraph D does not contain an outbranching 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 2 O(k log2 k) · n O(1) whether a strongly connected digraph on n vertices has an outbranching with at least k leaves. On acyclic digraphs the running time of our algorithm is 2 O(k log k) · n O(1). 1
An FPT Algorithm for Directed Spanning kLeaf
 Preprint 0462007, Combinatorial Optimization & Graph Algorithms Group
, 2007
"... An outbranching of a directed graph is a rooted spanning tree with all arcs directed outwards from the root. We consider the problem of deciding whether a given digraph D has an outbranching with at least k leaves (Directed Spanning kLeaf). We prove that this problem is fixed parameter tractable, ..."
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Cited by 5 (0 self)
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An outbranching of a directed graph is a rooted spanning tree with all arcs directed outwards from the root. We consider the problem of deciding whether a given digraph D has an outbranching with at least k leaves (Directed Spanning kLeaf). We prove that this problem is fixed parameter tractable, when k is chosen as the parameter. Previously this was only known for restricted classes of directed graphs. The main new ingredient in our approach is a lemma that shows that given a locally optimal outbranching of a directed graph in which every arc is part of at least one outbranching, either an outbranching with at least k leaves exists, or a path decomposition with width O(k 3) can be found. This enables a dynamic programming based algorithm of running time 2 O(k3 log k) · n O(1), where n = V (D).
Outbranchings with Extremal Number of Leaves
"... A subdigraph T of a digraph D is called an outtree if T is an oriented tree with just one vertex s of indegree zero. A spanning outtree is called an outbranching. A vertex x of an outbranching B is called a leaf if d + B (x) = 0. This is mainly a survey paper on outbranchings with minimum and ..."
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A subdigraph T of a digraph D is called an outtree if T is an oriented tree with just one vertex s of indegree zero. A spanning outtree is called an outbranching. A vertex x of an outbranching B is called a leaf if d + B (x) = 0. This is mainly a survey paper on outbranchings with minimum and maximum number of leaves. We give short proofs of some wellknown theorems. 1
Combinatorial Algorithms
"... We present an O ( √ opt)approximation algorithm for the maximum leaf spanning arborescence problem, where opt is the number of leaves in an optimal spanning arborescence. The result is based upon an O(1)approximation algorithm for a special class of directed graphs called willows. Incorporating t ..."
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We present an O ( √ opt)approximation algorithm for the maximum leaf spanning arborescence problem, where opt is the number of leaves in an optimal spanning arborescence. The result is based upon an O(1)approximation algorithm for a special class of directed graphs called willows. Incorporating the method for willow graphs as a subroutine in a local improvement algorithm gives the bound for general directed graphs.
On the Parameterized MaxLeaf Problems: Digraphs and Undirected Graphs ∗
"... The parameterized maxleaf problem on undirected graphs (which is also named the maxleaf spanningtree problem) is formulated as follows: given an undirected graph G and a parameter k, either construct a spanning tree with at least k leaves for G or report ‘No ’ if such a tree does not exist. The p ..."
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The parameterized maxleaf problem on undirected graphs (which is also named the maxleaf spanningtree problem) is formulated as follows: given an undirected graph G and a parameter k, either construct a spanning tree with at least k leaves for G or report ‘No ’ if such a tree does not exist. The problem also has a version for directed graphs that is named the maxleaf outbranching problem. In this paper, we present a simple branchandsearch algorithm of running time O ∗ (4 k) that solves the maxleaf outbranching problem. This significantly improves the previous best algorithm for the problem that runs in time O ∗ (2 O(k log k)). Our main contributions consist of new observations on the combinatorial structures of the problem and the introduction of a new algorithmic technique that provides new perspectives for design and analysis of parameterized algorithms. Our algorithm of running time O ∗ (4 k) is also applicable to the simpler maxleaf spanningtree problem, improving the previous best algorithm of running time O ∗ (6.75 k) for the problem. 1