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Bidimensionality and Kernels
, 2010
"... Bidimensionality theory appears to be a powerful framework in the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for bidimensional problems on Hminor free graphs. Demaine and Hajiaghayi ..."
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Cited by 21 (11 self)
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Bidimensionality theory appears to be a powerful framework in the development of metaalgorithmic techniques. It was introduced by Demaine et al. [J. ACM 2005] as a tool to obtain subexponential time parameterized algorithms for bidimensional problems on Hminor free graphs. Demaine and Hajiaghayi [SODA 2005] extended the theory to obtain polynomial time approximation schemes (PTASs) for bidimensional problems. In this paper, we establish a third metaalgorithmic direction for bidimensionality theory by relating it to the existence of linear kernels for parameterized problems. In parameterized complexity, each problem instance comes with a parameter k and the parameterized problem is said to admit a linear kernel if there is a polynomial time algorithm, called
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.
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
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 6 (5 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)
Minimum Leaf Outbranching and Related Problems
, 2008
"... 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 prove that MinLOB is polynomialtime solvable for acyclic digraphs. In general, MinLOB is NPhard and we co ..."
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Cited by 2 (0 self)
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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 prove that MinLOB is polynomialtime solvable for acyclic digraphs. In general, MinLOB is NPhard and we consider three parameterizations of MinLOB. We 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 2) and construct an algorithm of running time O(2 O(k log k) +n 6), which is an ‘additive ’ FPT algorithm. We also consider transformations from two related problems, the minimum path covering and the maximum internal outtree problems into MinLOB, which imply that some parameterizations of the two problems are FPT as well.
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
Directed Nowhere Dense Classes of Graphs
"... Many natural computational problems on graphs such as finding dominating or independent sets of a certain size are well known to be intractable, both in the classical sense as well as in the framework of parameterized complexity. Much work therefore has focussed on exhibiting restricted classes of g ..."
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Many natural computational problems on graphs such as finding dominating or independent sets of a certain size are well known to be intractable, both in the classical sense as well as in the framework of parameterized complexity. Much work therefore has focussed on exhibiting restricted classes of graphs on which these problems become tractable. While in the case of undirected graphs, there is a rich structure theory which can be used to develop tractable algorithms for these problems on large classes of undirected graphs, such a theory is much less developed for directed graphs. Many attempts to identify structure properties of directed graphs tailored towards algorithmic applications have focussed on a directed analogue of undirected treewidth. These attempts have proved to be successful in the development
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 ..."
Abstract
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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.