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A Partial KArboretum of Graphs With Bounded Treewidth
 J. Algorithms
, 1998
"... The notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies. This survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class. Also, some mutual relations between such classes ..."
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Cited by 264 (35 self)
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The notion of treewidth has seen to be a powerful vehicle for many graph algorithmic studies. This survey paper wants to give an overview of many classes of graphs that can be seen to have a uniform upper bound on the treewidth of graphs in the class. Also, some mutual relations between such classes are discussed.
The NPcompleteness column: an ongoing guide
 Journal of Algorithms
, 1985
"... This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & ..."
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Cited by 196 (0 self)
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This is the nineteenth edition of a (usually) quarterly column that covers new developments in the theory of NPcompleteness. The presentation is modeled on that used by M. R. Garey and myself in our book ‘‘Computers and Intractability: A Guide to the Theory of NPCompleteness,’ ’ W. H. Freeman & Co., New York, 1979 (hereinafter referred to as ‘‘[G&J]’’; previous columns will be referred to by their dates). A background equivalent to that provided by [G&J] is assumed, and, when appropriate, crossreferences will be given to that book and the list of problems (NPcomplete and harder) presented there. Readers who have results they would like mentioned (NPhardness, PSPACEhardness, polynomialtimesolvability, etc.) or open problems they would like publicized, should
Upper Bounds for Vertex Cover Further Improved
"... . The problem instance of Vertex Cover consists of an undirected graph G = (V; E) and a positive integer k, the question is whether there exists a subset C V of vertices such that each edge in E has at least one of its endpoints in C with jCj k. We improve two recent worst case upper bounds fo ..."
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Cited by 44 (17 self)
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. The problem instance of Vertex Cover consists of an undirected graph G = (V; E) and a positive integer k, the question is whether there exists a subset C V of vertices such that each edge in E has at least one of its endpoints in C with jCj k. We improve two recent worst case upper bounds for Vertex Cover. First, Balasubramanian et al. showed that Vertex Cover can be solved in time O(kn + 1:32472 k k 2 ), where n is the number of vertices in G. Afterwards, Downey et al. improved this to O(kn+ 1:31951 k k 2 ). Bringing the exponential base significantly below 1:3, we present the new upper bound O(kn + 1:29175 k k 2 ). 1 Introduction Vertex Cover is a problem of central importance in computer science: { It was among the rst NPcomplete problems [7]. { There have been numerous eorts to design ecient approximation algorithms [3], but it is also known to be hard to approximate [1]. { It is of central importance in parameterized complexity theory and has one ...
Scalable parallel algorithms for fpt problems
 Algorithmica
, 2006
"... Algorithmic methods based on the theory of fixedparameter tractability are combined with powerful computational platforms to launch systematic attacks on combinatorial problems of significance. As a case study, optimal solutions to very large instances of the NPhard vertex cover problem are comput ..."
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Cited by 25 (11 self)
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Algorithmic methods based on the theory of fixedparameter tractability are combined with powerful computational platforms to launch systematic attacks on combinatorial problems of significance. As a case study, optimal solutions to very large instances of the NPhard vertex cover problem are computed. To accomplish this, an efficient sequential algorithm and various forms of parallel algorithms are devised, implemented and compared. The importance of maintaining a balanced decomposition of the search space is shown to be critical to achieving scalability. Target problems need only be amenable to reduction and decomposition. Applications in high throughput computational biology are also discussed.
An Improved FixedParameterTractable Algorithm for Vertex Cover
, 1999
"... Given a graph G = (V; E), Vertex Cover asks for a smallest subset V 0 ` V such that for each edge (a; b) in G a 2 V 0 or b 2 V 0 . We present an improved fixedparameter tractable algorithm when the problem is parameterized by the size k of V 0 . The algorithm has a complexity of O(kn + max ..."
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Cited by 19 (5 self)
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Given a graph G = (V; E), Vertex Cover asks for a smallest subset V 0 ` V such that for each edge (a; b) in G a 2 V 0 or b 2 V 0 . We present an improved fixedparameter tractable algorithm when the problem is parameterized by the size k of V 0 . The algorithm has a complexity of O(kn + maxf(1:25542) k k 2 ; (1:2906) k 2:5kg). We improve the klam value by 16 to k = 157. 1 Introduction In 1972, Karp has shown that the following problem is NPcomplete [10]. Problem 1.1 Vertex Cover Instance: A graph G = (V; E), a positive integer k. Question: Does G have a vertex cover of size k? (I.e. does there exist a subset V 0 ` V , jV j k, such that for each (x; y) 2 E either x or y belongs to V 0 ?) Though NPcomplete, the following parameterized version was one of the first problems shown to be fixedparameter tractable [6, 9]. Problem 1.2 kVertex Cover Instance: A graph G = (V; E), a positive integer k. Parameter: k. Question: Does G have a vertex cover of size k?...
Optimal reduction of twoterminal directed acyclic graphs
 SIAM JOURNAL ON COMPUTING
, 1992
"... Algorithms for seriesparallel graphs can be extended to arbitrary twoterminal dags if node reductions are used along with series and parallel reductions. A node reduction contracts a vertex with unit indegree (outdegree) into its sole incoming (outgoing) neighbor. This paper gives an O(n2" ..."
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Cited by 16 (1 self)
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Algorithms for seriesparallel graphs can be extended to arbitrary twoterminal dags if node reductions are used along with series and parallel reductions. A node reduction contracts a vertex with unit indegree (outdegree) into its sole incoming (outgoing) neighbor. This paper gives an O(n2"5) algorithm for minimizing node reductions, based on vertex cover in a transitive auxiliary graph. Applications include the analysis of PERT networks, dynamic programming approaches to network problems, and network reliability. For NPhard problems one can obtain algorithms that are exponential only in the minimum number of node reductions rather than the number of vertices. This gives improvements if the underlying graph is nearly seriesparallel.
Computing Optimal Linear Layouts of Trees in Linear Time
 Proc. ESA 2000, number 1879
, 1999
"... We present a linear time algorithm which, given a tree, computes a linear layout optimal with respect to vertex separation. As a consequence optimal edge search strategies, optimal node search strategies, and optimal interval augmentations can be computed also in O(n) for trees. This improves the ru ..."
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Cited by 14 (0 self)
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We present a linear time algorithm which, given a tree, computes a linear layout optimal with respect to vertex separation. As a consequence optimal edge search strategies, optimal node search strategies, and optimal interval augmentations can be computed also in O(n) for trees. This improves the running time of former algorithms from O(n log n) to O(n) and answers two related open questions raised in [7] and [15].
Computing Excluded Minors
"... By Robertson and Seymour’s graph minor theorem, every minor ideal can be characterised by a finite family of excluded minors. (A minor ideal is a class of graphs closed under taking minors.) We study algorithms for computing excluded minor characterisations of minor ideals. We propose a general meth ..."
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Cited by 14 (4 self)
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By Robertson and Seymour’s graph minor theorem, every minor ideal can be characterised by a finite family of excluded minors. (A minor ideal is a class of graphs closed under taking minors.) We study algorithms for computing excluded minor characterisations of minor ideals. We propose a general method for obtaining such algorithms, which is based on definability in monadic secondorder logic and the decidability of the monadic secondorder theory of trees. A straightforward application of our method yields algorithms that, for a given k, compute excluded minor characterisations for the minor ideal Tk of all graphs of tree width at most k, the minor ideal Bk of all graphs of branch width at most k, and the minor ideal Gk of all graphs of genus at most k. Our main results are concerned with constructions of new minor ideals from given ones. Answering a question that goes back to Fellows and Langston [11], we prove that there is an algorithm that, given excluded minor characterisations of two minor ideals C and D, computes such a characterisation for the ideal C ∪ D. Furthermore, we obtain an algorithm for computing an excluded minor characterisation for the class of all apex graphs over a minor ideal C, given an excluded minor characterisation for C. (An apex graph over C is a graph G from which one vertex can be removed to obtain a graph in C.) A corollary of this result is a uniform ftpalgorithm for the “distance k from planarity” problem.
Scalable Parallel Algorithms for Difficult Combinatorial Problems: A Case Study
 In Proceedings, International Conference on Parallel and Distributed Computing and Systems (PDCS
, 2003
"... A novel combination of emergent algorithmic methods, powerful computational platforms and supporting infrastructure is described. These complementary tools and technologies are used to launch systematic attacks on combinatorial problems of significance. As a case study, optimal solutions to very lar ..."
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Cited by 13 (11 self)
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A novel combination of emergent algorithmic methods, powerful computational platforms and supporting infrastructure is described. These complementary tools and technologies are used to launch systematic attacks on combinatorial problems of significance. As a case study, optimal solutions to very large instances of the N Phard vertex cover problem are computed. To accomplish this, an efficient sequential algorithm and two forms of parallel algorithms are implemented. The importance of maintaining a balanced decomposition of the search space is shown to be critical to achieving scalability. With the synergistic combination of techniques detailed here, it is now possible to solve problem instances that before were widely viewed as hopelessly out of reach. Target problems need only be amenable to reduction and decomposition. Applications are also discussed.
Some Prospects for Efficient Fixed Parameter Algorithms
 In Proc. of the 25th Conference on Current Trends in Theory and Practice of Informatics (SOFSEM’98), Springer, LNCS 1521
, 1998
"... Recent time has seen quite some progress in the development of exponential time algorithms for NPhard problems, where the base of the exponential term is fairly small. These developments are also tightly related to the theory of fixed parameter tractability. In this incomplete survey, we explain so ..."
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Cited by 6 (0 self)
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Recent time has seen quite some progress in the development of exponential time algorithms for NPhard problems, where the base of the exponential term is fairly small. These developments are also tightly related to the theory of fixed parameter tractability. In this incomplete survey, we explain some basic techniques in the design of efficient fixed parameter algorithms, discuss deficiencies of parameterized complexity theory, and try to point out some future research challenges. The focus of this paper is on the design of efficient algorithms and not on a structural theory of parameterized complexity. Moreover, our emphasis will be laid on two exemplifying issues: Vertex Cover and MaxSat problems. A shorter version of this paper appears as an invited talk in the proceedings of the 25th Conference on Current Trends in Theory and Practice of Informatics (SOFSEM'98), Springer, LNCS , held in Jasna, Slovakia, November 2127, 1998. y Supported by a Feodor Lynen fellowship of the Alex...