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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 19 (7 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.
Beyond Bidimensionality: Parameterized Subexponential Algorithms on Directed Graphs
"... In 2000 Alber et al. [SWAT 2000] obtained the first parameterized subexponential algorithm on undirected planar graphs by showing that kDOMINATING SET is solvable in time 2 O( √ k) ..."
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Cited by 7 (6 self)
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In 2000 Alber et al. [SWAT 2000] obtained the first parameterized subexponential algorithm on undirected planar graphs by showing that kDOMINATING SET is solvable in time 2 O( √ k)
An exact algorithm for the maximum leaf spanning tree problem
 PROC. FOURTH INTERNATIONAL WORKSHOP ON PARAMETERIZED AND EXACT COMPUTATION. LECTURE NOTES IN COMPUTER SCIENCE
, 2009
"... 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 th ..."
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Cited by 6 (4 self)
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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 MeasureandConquer technique. Finally we provide a lower bound of Ω(1.4422 n) for the worst case running time of our algorithm.
Kernel(s) for Problems With No Kernel: On OutTrees With Many Leaves
, 2011
"... 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 5 (1 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 with O(k 3) vertices is obtained using extremal combinatorics. For the kLEAFOUTBRANCHING problem, we show that no polynomialsized kernel is possible unless coNP is in NP/poly. However, our positive results for ROOTED kLEAFOUTBRANCHING immediately imply that the seemingly intractable kLEAFOUTBRANCHING problem admits a data reduction to n independent polynomialsized kernels. These two results, tractability and intractability side by side, are the first ones separating Karp kernelization from Turing kernelization. This answers affirmatively an open problem
On the Directed DegreePreserving Spanning Tree Problem
"... Abstract. In this paper we initiate a systematic study of the Reduced Degree Spanning Tree problem, where given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree with at most k vertices of reduced outdegree. This problem is a directed analog of the wellstudied Mi ..."
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Abstract. In this paper we initiate a systematic study of the Reduced Degree Spanning Tree problem, where given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree with at most k vertices of reduced outdegree. This problem is a directed analog of the wellstudied MinimumVertex Feedback Edge Set problem. We show that this problem is fixedparameter tractable and admits a problem kernel with at most 8k vertices on strongly connected digraphs and O(k 2) vertices on general digraphs. We also give an algorithm for this problem on general digraphs with runtime O ∗ (5.942 k). This adds the Reduced Degree Spanning Tree problem to the small list of directed graph problems for which fixedparameter tractable algorithms are known. Finally, we consider the dual of Reduced Degree Spanning Tree, that is, given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree of D with at least k vertices of full outdegree. We show that this problem is W[1]hard on two important digraph classes: directed acyclic graphs and strongly connected digraphs. 1
On the Directed Full Degree Spanning Tree Problem
"... Abstract. We study the parameterized complexity of a directed analog of the Full Degree Spanning Tree problem where, given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree T of D such that at least k vertices in T have the same outdegree as in D. We show that t ..."
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Abstract. We study the parameterized complexity of a directed analog of the Full Degree Spanning Tree problem where, given a digraph D and a nonnegative integer k, the goal is to construct a spanning outtree T of D such that at least k vertices in T have the same outdegree as in D. We show that this problem is W[1]hard even on the class of directed acyclic graphs. In the dual version, called Reduced Degree Spanning Tree, one is required to construct a spanning outtree T such that at most k vertices in T have outdegrees that are different from that in D. We show that this problem is fixedparameter tractable and that it admits a problem kernel with at most 8k vertices on strongly connected digraphs and O(k 2 ) vertices on general digraphs. We also give an algorithm for this problem on general digraphs with running time O(5.942 k · n O(1) ), where n is the number of vertices in the input digraph.
kLEAF SPANNING TREE
"... The problem of finding a spanning tree in an undirected graph with a maximum number of leaves is known to be NPhard. We present an algorithm which finds a spanning tree with at least k leaves in time O∗(3.4575k) which improves the currently best algorithm. The estimation of the running time is done ..."
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The problem of finding a spanning tree in an undirected graph with a maximum number of leaves is known to be NPhard. We present an algorithm which finds a spanning tree with at least k leaves in time O∗(3.4575k) which improves the currently best algorithm. The estimation of the running time is done by using a nonstandard measure. The present paper is one of the still few examples that employ the Measure & Conquer paradigm of algorithm analysis in the area of Parameterized Algorithmics.