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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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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
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
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
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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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
BEYOND BIDIMENSIONALITY: PARAMETERIZED SUBEXPONENTIAL ALGORITHMS ON DIRECTED GRAPHS
, 2010
"... Abstract. In this paper we make the first step beyond bidimensionality by obtaining subexponential time algorithms for problems on directed graphs. We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approa ..."
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Abstract. In this paper we make the first step beyond bidimensionality by obtaining subexponential time algorithms for problems on directed graphs. We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, kLeaf OutBranching, which is to find an oriented spanning tree with at least k leaves, we obtain an algorithm solving the problem in time 2 O( √ k log k) n + n O(1) on directed graphs whose underlying undirected graph excludes some fixed graph H as a minor. For the special case when the input directed graph is planar, the running time can be improved to 2 O( √ k) n+n O(1). The second example is a generalization of the Directed Hamiltonian Path problem, namely kInternal OutBranching, which is to find an oriented spanning tree with at least k internal vertices. We obtain an algorithm solving the problem in time 2 O( √ k log k) + n O(1) on directed graphs whose underlying undirected graph excludes some fixed apex graph H as a minor. Finally, we observe that for any ε> 0, the kDirected Path problem is solvable in time O((1+ε) k n f(ε)), where f is some function of ε. Our methods are based on nontrivial combinations of obstruction theorems for undirected graphs, kernelization, problem specific combinatorial structures and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs. 1.
BEYOND BIDIMENSIONALITY: PARAMETERIZED SUBEXPONENTIAL ALGORITHMS ON DIRECTED GRAPHS
"... Abstract. In this paper we make the first step beyond bidimensionality by obtaining subexponential time algorithms for problems on directed graphs. We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approa ..."
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Abstract. In this paper we make the first step beyond bidimensionality by obtaining subexponential time algorithms for problems on directed graphs. We develop two different methods to achieve subexponential time parameterized algorithms for problems on sparse directed graphs. We exemplify our approaches with two well studied problems. For the first problem, kLeaf OutBranching, which is to find an oriented spanning tree with at least k leaves, we obtain an algorithm solving the problem in time 2 O( √ k log k) n + n O(1) on directed graphs whose underlying undirected graph excludes some fixed graph H as a minor. For the special case when the input directed graph is planar, the running time can be improved to 2 O( √ k) n+n O(1). The second example is a generalization of the Directed Hamiltonian Path problem, namely kInternal OutBranching, which is to find an oriented spanning tree with at least k internal vertices. We obtain an algorithm solving the problem in time 2 O( √ k log k) + n O(1) on directed graphs whose underlying undirected graph excludes some fixed apex graph H as a minor. Finally, we observe that for any ε> 0, the kDirected Path problem is solvable in time O((1+ε) k n f(ε)), where f is some function of ε. Our methods are based on nontrivial combinations of obstruction theorems for undirected graphs, kernelization, problem specific combinatorial structures and a layering technique similar to the one employed by Baker to obtain PTAS for planar graphs. 1.