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Width parameters beyond treewidth and their applications
 Computer Journal
, 2007
"... Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compare ..."
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Besides the very successful concept of treewidth (see [Bodlaender, H. and Koster, A. (2007) Combinatorial optimisation on graphs of bounded treewidth. These are special issues on Parameterized Complexity]), many concepts and parameters measuring the similarity or dissimilarity of structures compared to trees have been born and studied over the past years. These concepts and parameters have proved to be useful tools in many applications, especially in the design of efficient algorithms. Our presented novel look at the contemporary developments of these ‘width ’ parameters in combinatorial structures delivers—besides traditional treewidth and derived dynamic programming schemes—also a number of other useful parameters like branchwidth, rankwidth (cliquewidth) or hypertreewidth. In this contribution, we demonstrate how ‘width ’ parameters of graphs and generalized structures (such as matroids or hypergraphs), can be used to improve the design of parameterized algorithms and the structural analysis in other applications on an abstract level.
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 connecti ..."
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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.
Treewidth Lower Bounds with Brambles
, 2005
"... In this paper we present a new technique for computing lower bounds for graph treewidth. Our technique is based on the fact that the treewidth of a graph G is the maximum order of a bramble of G minus one. We give two algorithms: one for general graphs, and one for planar graphs. The algorithm fo ..."
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In this paper we present a new technique for computing lower bounds for graph treewidth. Our technique is based on the fact that the treewidth of a graph G is the maximum order of a bramble of G minus one. We give two algorithms: one for general graphs, and one for planar graphs. The algorithm for planar graphs is shown to give a lower bound for both the treewidth and branchwidth that is at most a constant factor away from the optimum. For both algorithms, we report on extensive computational experiments that show that the algorithms give often excellent lower bounds, in particular when applied to (close to) planar graphs.
Computing branch decomposition of large planar graphs
 In Experimental Algorithms
, 2008
"... Abstract: A graph of small branchwidth admits efficient dynamic programming algorithms for many NPhard problems on the graph. A key step in these algorithms is to find a branch decomposition of small width for the graph. Given a planar graph G of n vertices, an optimal branch decomposition of G ca ..."
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Abstract: A graph of small branchwidth admits efficient dynamic programming algorithms for many NPhard problems on the graph. A key step in these algorithms is to find a branch decomposition of small width for the graph. Given a planar graph G of n vertices, an optimal branch decomposition of G can be computed in polynomial time, e.g., by the edgecontraction method in O(n3) time. All known algorithms for the planar branch decomposition use Seymour and Thomas procedure which, given an integer β, decides whether G has the branchwidth at least β or not in O(n2) time. Recent studies report efficient implementations of Seymour and Thomas procedure that compute the branchwidth of planar graphs of size up to one hundred thousand edges in a practical time and memory space. Using the efficient implementations as a subroutine, it is reported that the edgecontraction method computes an optimal branch decomposition for planar graphs of size up to several thousands edges in a practical time but it is still time consuming for graphs with larger size. In this paper, we propose divideandconquer based algorithms of using Seymour and Thomas procedure to compute optimal branch decompositions of planar graphs. Our algorithms have time complexity O(n3). Computational studies show that our algorithms are much faster than the edgecontraction algorithms and can compute an optimal branch decomposition of some planar graphs of size up to 50,000 edges in a practical time.
A combinatorial optimization algorithm for solving the branchwidth problem
 Rice University
, 2010
"... In this paper, we consider the problem of computing an optimal branch decomposition of a graph. Branch decompositions and branchwidth were introduced by Robertson and Seymour in their series of papers that proved the Graph Minors Theorem. Branch decompositions have proven to be useful in solving man ..."
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In this paper, we consider the problem of computing an optimal branch decomposition of a graph. Branch decompositions and branchwidth were introduced by Robertson and Seymour in their series of papers that proved the Graph Minors Theorem. Branch decompositions have proven to be useful in solving many NPhard problems, such as the traveling salesman, independent set, and ring routing problems, by means of combinatorial algorithms that operate on branch decompositions. We develop an implicit enumeration algorithm for the optimal branch decomposition problem and examine its performance on a set of classical graph instances.
Graphs, Branchwidth, and Tangles! Oh My!
"... Branch decompositionbased algorithms have been used in practical settings to solve some NPhard problems like the travelling salesman problem (TSP) and general minor containment. The notions of branch decompositions and branchwidth were introduced by Robertson and Seymour to assist in proving the G ..."
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Branch decompositionbased algorithms have been used in practical settings to solve some NPhard problems like the travelling salesman problem (TSP) and general minor containment. The notions of branch decompositions and branchwidth were introduced by Robertson and Seymour to assist in proving the Graph Minors Theorem. Given a connected graph G and a branch decomposition of G of width k where k is at least 3, a practical branch decompositionbased algorithm to test whether a graph has branchwidth at most k − 1is given. The algorithm either constructs a branch decomposition of G of width at most k − 1 or constructs a tangle basis of order k, which offers a lower bound on the branchwidth of G. The algorithm is utilized repeatedly in a practical setting to find an optimal branch decomposition of a connected graph, whose branchwidth is at least 2, given an input branch decomposition of the graph from a heuristic. This is the first algorithm for the optimal branch decomposition problem for general graphs that has been shown to be practical. Computational results are provided to illustrate the effectiveness of finding optimal branch decompositions. A tangle basis is related to a tangle, a notion also introduced by Robertson and Seymour; however, a tangle basis is more constructive in nature. Furthermore, it is shown that a tangle basis of order k is coextensive to a tangle of order k. © 2004 Wiley Periodicals, Inc. NETWORKS, Vol. 45(2), 55–60 2005 Keywords: basis branchwidth; branch decomposition; tangle; tangle 1.
Tree/Branch Decompositions and Their Applications
"... The notions of tree decompositions and branch decompositions have received much attention in discrete optimizations. These notions were originally introduced by Robertson and Seymour [15, 16] in the proof of the Graph Minors Theorem, known as Wagner’s conjecture. It is known that practical problems ..."
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The notions of tree decompositions and branch decompositions have received much attention in discrete optimizations. These notions were originally introduced by Robertson and Seymour [15, 16] in the proof of the Graph Minors Theorem, known as Wagner’s conjecture. It is known that practical problems in several research areas, like VLSI design, Cholesky factorization, evolution theory, control flow graphs of computer programs, and 0/1 integer programs, are equivalent to treewidth of an associated graph. Overviews on these applications can be found in Bodlaender [5] and Hicks et al. [13]. Courcelle [9] and Arnborg et al. [3] showed that several NPhard problems on graphs can be solved in polynomial time using dynamic programming techniques if the graphs have bounded treewidth or branchwidth. There are two major steps in the techniques of [9, 3]: (1) compute a tree/branch decomposition with small treewidth/branchwidth of the input graph; and (2) use a dynamic programming based algorithm to solve the problem with the help of the tree/branch decomposition found in step (1). These techniques are known as tree/branch decomposition based algorithms and revealed the algorithmic importance of tree decompositions and branch decompositions. Efficient algorithms for instances with bounded treewidth/branchwidth of many NPhard problems have been developed based these techniques. You are referred to Bodlaender and Hicks et al. for overviews [5, 13] • Terminology In this note, a graph is undirected unless otherwise stated, and is simple (without parallel edges and selfloops), unless it is explicitly referred as a multigraph. For a graph or multigraph G, we denote by V (G) the set of vertices of G and by E(G) the set of edges of G. For each X ⊆ V (G), we denote by E(X) the set of edges incident with two vertices in X and by δ(X) the set of edges incident with one vertex in X and one vertex in V (G) \ X. For disjoint X, Y ⊆ V (G), we denote by E(X, Y) the set of edges incident with one vertex in X and one vertex in Y. For each E ′ ⊆ E(G), we denote by V (E ′ ) the set of end vertices of the edges in E ′. • Tree decompositions A treedecomposition of a given graph G(V, E) is a pair (f, T), where T is a tree and f associates each node p of T a subset f(p) ⊆ V (G) such that
On the minimum corridor connection problem and other generalized geometric problems
"... On the minimum corridor connection problem and other generalized geometric problems *# Hans Bodlaender# # Corinne Feremans $ Alexander Grigoriev ..."
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On the minimum corridor connection problem and other generalized geometric problems *# Hans Bodlaender# # Corinne Feremans $ Alexander Grigoriev