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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 24 (13 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 9 (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.
Algorithm for Finding kVertex Outtrees and its Application to kInternal Outbranching Problem
 In COCOON (2009), 37–46
"... An outtree T is an oriented tree with only one vertex of indegree zero. A vertex x of T is internal if its outdegree is positive. We design randomized and deterministic algorithms for deciding whether an input digraph contains a given outtree with k vertices. The algorithms are of runtime O ∗ (5 ..."
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Cited by 8 (6 self)
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An outtree T is an oriented tree with only one vertex of indegree zero. A vertex x of T is internal if its outdegree is positive. We design randomized and deterministic algorithms for deciding whether an input digraph contains a given outtree with k vertices. The algorithms are of runtime O ∗ (5.704 k) and O ∗ (5.704 k(1+o(1))), respectively. We apply the deterministic algorithm to obtain a deterministic algorithm of runtime O ∗ (c k), where c is a constant, for deciding whether an input digraph contains a spanning outtree with at least k internal vertices. This answers in affirmative a question of Gutin, Razgon and Kim (Proc. AAIM’08). 1
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)
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
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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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.
Directed Nowhere Dense Classes of Graphs
, 2012
"... 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
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
Cliquewidth: When Hard Does Not Mean Impossible∗
"... In recent years, the parameterized complexity approach has lead to the introduction of many new algorithms and frameworks on graphs and digraphs of bounded cliquewidth and, equivalently, rankwidth. However, despite intensive work on the subject, there still exist wellestablished hard problems whe ..."
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In recent years, the parameterized complexity approach has lead to the introduction of many new algorithms and frameworks on graphs and digraphs of bounded cliquewidth and, equivalently, rankwidth. However, despite intensive work on the subject, there still exist wellestablished hard problems where neither a parameterized algorithm nor a theoretical obstacle to its existence are known. Our article is interested mainly in the digraph case, targeting the wellknown Minimum Leaf OutBranching (cf. also Minimum Leaf Spanning Tree) and Edge Disjoint Paths problems on digraphs of bounded cliquewidth with nonstandard new approaches. The first part of the article deals with the Minimum Leaf OutBranching problem and introduces a novel XPtime algorithm wrt. cliquewidth. We remark that this problem is known to be W[2]hard, and that our algorithm does not resemble any of the previously published attempts solving special cases of it such as the Hamiltonian Path. The second part then looks at the Edge Disjoint Paths problem (both on graphs and digraphs) from a different perspective – rather surprisingly showing that this problem has a definition in the MSO1 logic of graphs. The lineartime FPT algorithm wrt. cliquewidth then follows as a direct consequence.
Digraph Width Measures in Parameterized Algorithmics
"... In contrast to undirected width measures such as treewidth, which have provided many important algorithmic applications, analogous measures for digraphs such as directed treewidth or DAGwidth do not seem so successful. Several recent papers have given some evidence on the negative side. We conf ..."
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In contrast to undirected width measures such as treewidth, which have provided many important algorithmic applications, analogous measures for digraphs such as directed treewidth or DAGwidth do not seem so successful. Several recent papers have given some evidence on the negative side. We confirm and consolidate this overall picture by thoroughly and exhaustively studying the complexity of a range of directed problems with respect to various parameters, and by showing that they often remain NPhard even on graph classes that are restricted very beyond having small DAGwidth. On the positive side, it turns out that cliquewidth (of digraphs) performs much better on virtually all considered problems, from the parameterized complexity point of view.