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26
Efficient and Constructive Algorithms for the Pathwidth and Treewidth of Graphs
, 1993
"... In this paper we give, for all constants k, l, explicit algorithms, that given a graph G = (V; E) with a treedecomposition of G with treewidth at most l, decide whether the treewidth (or pathwidth) of G is at most k, and if so, find a treedecomposition or (pathdecomposition) of G of width at most ..."
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Cited by 75 (13 self)
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In this paper we give, for all constants k, l, explicit algorithms, that given a graph G = (V; E) with a treedecomposition of G with treewidth at most l, decide whether the treewidth (or pathwidth) of G is at most k, and if so, find a treedecomposition or (pathdecomposition) of G of width at most k, and that use O(V) time. In contrast with previous solutions, our algorithms do not rely on nonconstructive reasoning, and are single exponential in k and l. This result can be combined with a result of Reed [37], yielding explicit O(n log n) algorithms for the problem, given a graph G, to determine whether the treewidth (or pathwidth) of G is at most k, and if so, to find a tree (or path)decomposition of width at most k (k constant). Also, Bodlaender [13] has used the result of this paper to obtain linear time algorithms for these problems. We also show that for all constants k, there exists a polynomial time algorithm, that, when given a graph G = (V; E) with treewidth k, computes the pathwidth of G and a minimum path decomposition of G.
Beyond NPCompleteness for Problems of Bounded Width: Hardness for the W Hierarchy (Extended Abstract)
 In Proceedings of the 26th Annual ACM Symposium on the Theory of Computing
, 1994
"... The parameterized computational complexity of a collection of wellknown problems including: Bandwidth, Precedence constrained kprocessor scheduling, Longest Common Subsequence, DNA physical mapping (or Intervalizing colored graphs), Perfect phylogeny (or Triangulating colored graphs), Colored cutw ..."
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Cited by 60 (20 self)
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The parameterized computational complexity of a collection of wellknown problems including: Bandwidth, Precedence constrained kprocessor scheduling, Longest Common Subsequence, DNA physical mapping (or Intervalizing colored graphs), Perfect phylogeny (or Triangulating colored graphs), Colored cutwidth, and Feasible register assignment is explored. It is shown that these problems are hard for various levels of the W hierarchy. In the case of Precedence constrained kprocessor scheduling the results can be interpreted as providing substantial new complexity lower bounds on the outcome of [OPEN 8] of the Garey and Johnson list. We also obtain the conjectured "third strike" against Perfect phylogeny.
Constructive Linear Time Algorithms for Branchwidth
, 1997
"... We prove that, for any fixed k, one can construct a linear time algorithm that checks if a graph has branchwidth k and, if so, outputs a branch decomposition of minimum width. 1 Introduction This paper considers the problem of finding branch decompositions of graphs with small branchwidth. The noti ..."
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Cited by 33 (7 self)
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We prove that, for any fixed k, one can construct a linear time algorithm that checks if a graph has branchwidth k and, if so, outputs a branch decomposition of minimum width. 1 Introduction This paper considers the problem of finding branch decompositions of graphs with small branchwidth. The notion of branchwidth has a close relationship to the more wellknown notion of treewidth, a notion that has come to play a large role in many recent investigations in algorithmic graph theory. (See Section 2 for definitions of treewidth and branchwidth.) One reason for the interest in this notion is that many graph problems can be solved by linear time algorithms, when the inputs are restricted to graphs with some uniform upper bound on their treewidth. Most of these algorithms first try to find a tree decomposition of small width, and then utilize the advantages of the tree structure of the decomposition (see [1], [4]). The branchwidth of a graph differs from its treewidth by at most a multipl...
Approximation Algorithms for Classes of Graphs Excluding SingleCrossing Graphs as Minors
"... Many problems that are intractable for general graphs allow polynomialtime solutions for structured classes of graphs, such as planar graphs and graphs of bounded treewidth. ..."
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Cited by 26 (18 self)
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Many problems that are intractable for general graphs allow polynomialtime solutions for structured classes of graphs, such as planar graphs and graphs of bounded treewidth.
Graphs with Branchwidth at most Three
 J. Algorithms
, 1997
"... In this paper we investigate both the structure of graphs with branchwidth at most three, as well as algorithms to recognise such graphs. We show that a graph has branchwidth at most three, if and only if it has treewidth at most three and does not contain the threedimensional binary cube graph as ..."
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Cited by 23 (2 self)
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In this paper we investigate both the structure of graphs with branchwidth at most three, as well as algorithms to recognise such graphs. We show that a graph has branchwidth at most three, if and only if it has treewidth at most three and does not contain the threedimensional binary cube graph as a minor. A set of four graphs is shown to be the obstruction set of graphs with branchwidth at most three. We give a safe and complete set of reduction rules for the graphs with branchwidth at most three. Using this set, a linear time algorithm is given that checks if a given graph has branchwidth at most three, and, if so, outputs a minimum width branch decomposition. Keywords: graph algorithms, branchwidth, obstruction set, graph minor, reduction rule. 1 Introduction This paper considers the study of the graphs with branchwidth at most three. The notion of branchwidth has a close relationship to the more wellknown notion of treewidth, a notion that has come to play a large role in many ...
Branch and Tree Decomposition Techniques for Discrete Optimization
, 2005
"... This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectiv ..."
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Cited by 21 (3 self)
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This chapter gives a general overview of two emerging techniques for discrete optimization that have footholds in mathematics, computer science, and operations research: branch decompositions and tree decompositions. Branch decompositions and tree decompositions along with their respective connectivity invariants, branchwidth and treewidth, were first introduced to aid in proving the Graph Minors Theorem, a wellknown conjecture (Wagner’s conjecture) in graph theory. The algorithmic importance of branch decompositions and tree decompositions for solving NPhard problems modelled on graphs was first realized by computer scientists in relation to formulating graph problems in monadic second order logic. The dynamic programming techniques utilizing branch decompositions and tree decompositions, called branch decomposition and tree decomposition based algorithms, fall into a class of algorithms known as fixedparameter tractable algorithms and have been shown to be effective in a practical setting for NPhard problems such as minimum domination, the travelling salesman problem, general minor containment, and frequency assignment problems.
Safe Reduction Rules For Weighted Treewidth
, 2002
"... Several sets of reductions rules are known for preprocessing a graph when computing its treewidth. In this paper, we give reduction rules for a weighted variant of treewidth, motivated by the analysis of algorithms for probabilistic networks. We present two general reduction rules that are safe for ..."
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Cited by 15 (6 self)
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Several sets of reductions rules are known for preprocessing a graph when computing its treewidth. In this paper, we give reduction rules for a weighted variant of treewidth, motivated by the analysis of algorithms for probabilistic networks. We present two general reduction rules that are safe for weighted treewidth. They generalise many of the existing reduction rules for treewidth. Experimental results show that these reduction rules can significantly reduce the problem size for several instances of reallife probabilistic networks.
Subgraph Isomorphism, logBounded Fragmentation, and Graphs of (Locally) Bounded Treewidth
 in Proc. the 27th International Symposium on Mathematical Foundations of Computer Science
, 2002
"... The subgraph isomorphism problem, that of nding a copy of one graph in another, has proved to be intractable except when certain restrictions are placed on the inputs. In this paper, we introduce a new property for graphs (a generalization on bounded degree) and extend the known classes of input ..."
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Cited by 13 (4 self)
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The subgraph isomorphism problem, that of nding a copy of one graph in another, has proved to be intractable except when certain restrictions are placed on the inputs. In this paper, we introduce a new property for graphs (a generalization on bounded degree) and extend the known classes of inputs for which polynomialtime subgraph isomorphism algorithms are attainable. In particular, if the removal of any set of at most k vertices from an nvertex graph results in O(k log n) connected components, we say that the graph is a logbounded fragmentation graph. We present a polynomialtime algorithm for nding a subgraph of H isomorphic to a graph G when G is a logbounded fragmentation graph and H has bounded treewidth; these results are extended to handle graphs of locally bounded treewidth (a generalization of treewidth) when G is a logbounded fragmentation graph and has constant diameter.
Partitioning Problems: Characterization, Complexity and Algorithms on Partial kTrees
, 1994
"... This thesis investigates the computational complexity of algorithmic problems defined on graphs. At the abstract level of the complexity spectrum we discriminate polynomialtime solvable problems from N Pcomplete problems, while at the concrete level we improve on polynomialtime algorithms for gen ..."
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Cited by 8 (1 self)
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This thesis investigates the computational complexity of algorithmic problems defined on graphs. At the abstract level of the complexity spectrum we discriminate polynomialtime solvable problems from N Pcomplete problems, while at the concrete level we improve on polynomialtime algorithms for generally hard problems restricted to treedecomposable graphs. One contribution of this thesis is a precise characterization of vertex partitioning problems which include variants of domination, coloring and packing. An elaboration of this characterization is given for problems defined over vertex subsets and over maximal/minimal vertex subsets. We introduce several new graph parameters as vertex partition generalizations of classical parameters. The given characterizations provide a basis for a taxonomy of vertex partitioning problems, facilitating their common algorithmic treatment and allowing for their uniform complexity classification. We explore the computational complexity of two important types of problems
PreProcessing Rules for Triangulation of Probabilistic Networks
 COMPUTATIONAL INTELLIGENCE
, 2003
"... The currently most efficient algorithm for inference with a probabilistic network builds upon a triangulation of a network's graph. In this paper, we show that preprocessing can help in finding good triangulations for probabilistic networks, that is, triangulations with a minimal maximum cliqu ..."
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Cited by 7 (5 self)
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The currently most efficient algorithm for inference with a probabilistic network builds upon a triangulation of a network's graph. In this paper, we show that preprocessing can help in finding good triangulations for probabilistic networks, that is, triangulations with a minimal maximum clique size. We provide a set of rules for stepwise reducing a graph, without losing optimality. This reduction allows us to solve the triangulation problem on a smaller graph. From the smaller graph's triangulation, a triangulation of the original graph is obtained by reversing the reduction steps. Our experimental results show that the graphs of some wellknown reallife probabilistic networks can be triangulated optimally just by preprocessing; for other networks, huge reductions in their graph's size are obtained.