H. Bodlaender, R. J. Gilbert, H. Hafsteinson, T. Kloks, Approximating treewidth, pathwidth, frontsize, and shortest elimination tree, J. Algorithms 18 (1995) 238-255.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....has tree width n, and a complete graph has tree width n 1. Determining the tree width of a graph is NP hard [10] However, recognizing graphs of tree width at most k is linear for k = O(1) and there is an O(log n) approximation algorithm for computing the tree width of an arbitrary graph (see [17] and also [16, 31] We show the following general lower bound: end communication problem uses headers of size at least t 3 log ) bits where is the tree width of the graph G s;t obtained from G, s, and t by deleting every edge e not on a simple path from s to t. As we will see in more ....

H. Bodlaender, J. Gilbert, H. Hafsteinsson, and T. Kloks. Approximating Treewidth, Pathwidth, Frontsize, and Shortest Elimination Tree. Journal of Algorithms 18:238-255 (1995).

.... other hand, treewidth can be computed exactly in polynomial time for chordal graphs, permutation graphs [BKK95] circular arc graphs [SSR94] circle graphs [Klo96] distance hereditary graphs [BDK00] and for graphs of a xed treewidth [Bod96] From the approximation viewpoint, Bodlaender et al. [BGHK95] gave an O(log n) approximation algorithm for treewidth on general graphs. A famous open problem is whether treewidth can be approximated within a constant factor. Constant factor approximations are possible on AT free graphs [BT01,BKMT01] and on planar graphs. The approximation for planar graphs ....

Hans L. Bodlaender, John R. Gilbert, Hjalmtyr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms, 18(2):238-255, 1995.

.... degree at most nine, bipartite graphs and cocomparability graphs [BT97] This problem has been solved for several classes of graphs such as chordal graphs, permutation graphs [BKK95] circular arc graphs [SSR94] circle graphs [Klo93] and distance hereditary graphs [BDK00] Bodlaender et al. [BGHK95] gave an approximation algorithm with performance ratio O(logn) for this problem on general graphs. Solving the problem for the case in which the parameter k is fixed is also interesting. The first polynomial time algorithm for this problem was presented by Arnborg, Corneil and Proskurowski ....

Hans L. Bodlaender, John R. Gilbert, Hj'almt'yr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms, 18(2):238--255, 1995.

.... of bipartite graphs [ACP87] On the other hand, treewidth can be computed exactly in polynomial time for chordal graphs, permutation graphs [BKK95] circular arc graphs [SSR94] circle graphs [Klo93] and distance hereditary graphs [BDK00] From the approximation viewpoint, Bodlaender et al. [BGHK95] gave an O(log n) approximation algorithm for treewidth on general graphs. A famous open problem is whether treewidth can be approximated within constant factor. Treewidth can be approximated within constant factor on AT free graphs [BT01] see also [BKMT] and on planar graphs. The approximation ....

Hans L. Bodlaender, John R. Gilbert, Hjalmtyr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms, 18(2):238-255, 1995.

.... degree at most nine, bipartite graphs and cocomparability graphs [BT97] This problem has been solved for several classes of graphs such as chordal graphs, permutation graphs [BKK95] circular arc graphs [SSR94] circle graphs [Klo93] and distance hereditary graphs [BDK00] Bodlaender et al. BGHK95] gave an approximation algorithm with performance ratio O(log n) for this problem on general graphs. Solving the problem for the case in which the parameter k is fixed is also interesting. The first polynomial time algorithm for this problem was presented by Arnborg, Corneil and Proskurowski ....

Hans L. Bodlaender, John R. Gilbert, Hjalmtyr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms, 18(2):238--255, 1995.

....of W # X , making a clique of G[W # X] Figure 1: Triangulate G such that W is a clique. 4 The algorithm uses the simple principle of divide and conquer. The main improvement in this algorithm over similar previous algorithms (e.g. the O(k log n) factor approximation algorithm of [24, 10]) is in using an exact 1 2 vertex separator for W in G. The computational efficiency of the algorithm relies on an efficient way to find a 1 2 vertex separator for W in G, which we provide below. It is also important to notice that there is no benefit in finding a 1 2 vertex separator for V ....

Hans L. Bodlaender, John R. Gilbert, Hjalmtyr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. Journal of Algorithms, 18(2):238--255, March 1995.

....these parameters, we try to exhaustively review existing parameters and the relations that may exist between then. In the process we exhibit some missing relations. Several existing results, both old results and recent results from graph theory and Cholesky matrix factorization technology [BGHK95] allow us to give a very dense map of relations between these parameters. These results strongly relate several existing algorithms and answer some questions which were considered as open in the CSP community. Warning: this document is a working paper. Some sections may be incomplete or ....

....over all tree decompositions of G. A path decomposition of G is a tree decomposition (fX i g; T ) such that T is a path. The pathwidth of such a path decomposition is max i jX i j 1. The pathwidth of G is the minimum pathwidth over all path decompositions of G. Definition 10 (Separator number [BGHK95] Let be a constant between 0 and 1. An vertex separator of G is a set S V of vertices such that every connected component of the graph induced by V S has at most :jV j vertices. For any W V , an vertex separator of W in G is a set S V of vertices such that every connected component ....

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Hans L. Bodlaender, John R. Gilbert, Hjlmtr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. Journal of Algorithms, 18:238--255, 1995.

....are mapped to a path. The problems of minimizing pathwidth or cutwidth of a graph are interesting in some VLSI gate layout problems, see [43, 52] The problem also appears in minimizing vertex or edge rankings of graphs which are used in manufacturing systems [45] see [6] for more details) In [30, 1, 5] an O(log 2 n) approximation was presented. Planar Completion: The Minimum Drawing Size is de ned as : given a graph G, to provide a planar embedding mapping vertices to points and edges to simple curves, such that the total number 1 of points are minimized. At most two edges can intersect at ....

H. L. Bodlaender, J. R. Gilbert,H. Hafsteinsson, and T. Kloks. \Approximating treewidth, pathwidth, frontsize, and shortest elimination tree". In Jornal of Algorithms v18(2), pages 238-255, (1995)

....(see Section 5) has arity 2. Therefore if it is established that computing the treewidth of a graph is hard to approximate within a factor larger than 2, it would imply NP hardness of computing query width. The best known approximation algorithm for computing treewidth has a ratio of O(log n) [5], so the above possibility is not ruled out. In addition, even if computing query width is hard, we believe that finding a query decomposition of size k, if it exists, can be computed in polynomial time for each fixed k. Our containment algorithms are polynomial only for fixed width queries, hence ....

H. Bodlaender, J. Gilbert, H. Hafsteinsson, and T. Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. Journal of Algorithms, 18(2):238--255, March 1995.

....the control flow graphs of their programs have simple listings bounded tree width. For variable k, the problem of deciding the tree width is NPhard [4] For fixed k, however, there are linear time algorithms [8] Also, for variable k, there has been work done on polynomial approximating algorithms [10]. Moreover, from Bellcore, there is a commercially available tree width heuristic by Cook and Seymour. Our derivation of simple listings from syntax, as described in Theorem 3, is, however, much simpler than the general approaches to tree width, so the general advice following from this paper is: ....

H.L. Bodlaender, J.R. Gilbert, H. Hafsteinsson, and T. Kloks, Approximating Treewidth, Pathwidth, Frontsize, and Shortest Elimination Tree, J. Algorithms 18, 2 (1995) 221--237.

....generated. At the end of each iteration, the labels stored in the array B define a partial labeling. The positive labels in use always form a subset of the labels used by an optimal labeling (see Section 4) That means, the procedure never uses a wrong label. On exit from the while loop, B[1] B[2]; B[d] are all positive and form a valid labeling. In Section 4 we give a characterization of this valid labeling and prove that it is actually optimal. Based on the procedure labeling, we can construct an algorithm for computing a critical edge ranking of a rooted tree R. For each node v ....

....ranking of a rooted tree R. For each node v in R, let R v denote the subtree in R rooted at v. We execute labeling(R v ) for each internal node v of R in a bottom up manner. The input to labeling(R v ) consists of the critical sets of the subtrees rooted Procedure labeling(T ) Input: L[1] L[2]; L[d] the critical sets of T 1 ; T 2 ; Delta Delta Delta ; T d ) Output: B[1] B[d] the branch labels) and L (the critical set of T ) ffi for i = 1; 2; d do f B[i] 0; b L[i] L[i] g unlabeled branch : d; K : empty set; Step I Initialization ffi while ....

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H. L. Bodlaender, J. R. Gilbert, H. Hafsteinsson, and T. Kloks, Approximating treewidth, pathwidth, frontsize, and shortest elimination tree, J. Algorithms, 18 (1995), pp. 238-255.

....graph, and focus our efforts on finding parallel elimination orders with linear fill. 1. 4 Related work: Height Ignoring fill, computing an elimination ordering for a given graph with minimum height is NP hard [30] and remains so even if an additive error in the estimate of the height is allowed [5]. Pan and Reif give one of the first analyses of the parallel height of nested dissection orderings as well as how nested dissection can be used for solving the shortest path problem in graphs [28, 27] Bodlaender et al. 5] uses an approach similar to [1] to find elimination orders with bounds on ....

.... so even if an additive error in the estimate of the height is allowed [5] Pan and Reif give one of the first analyses of the parallel height of nested dissection orderings as well as how nested dissection can be used for solving the shortest path problem in graphs [28, 27] Bodlaender et al. [5] uses an approach similar to [1] to find elimination orders with bounds on the height and several related parameters. Both these papers [1, 5] give elimination orders with height at most O(log 2 n) times the minimum possible, for any n node graph. Numerous heuristics without performance ....

[Article contains additional citation context not shown here]

Hans L. Bodlaender, John R. Gilbert, Hj'almt'yr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. Journal of Algorithms, 18(2):238--255, March 1995.

....control flow graphs of their programs have simple listings bounded tree width. For variable k, the problem of deciding the tree width is NP hard [3] For fixed k, however, there are linear time algorithms [8] Also, for variable k, there has been work done on polynomial approximating algorithms [10]. Finally, the heuristic presented in Appendix A may be of some help, as it deals directly with three address code that may contain any number of programmer supplied gotos. Our derivation of simple listings from syntax, as described in Section 2, is, however, much simpler than the general ....

H.L. Bodlaender, J.R. Gilbert, H. Hafsteinsson, and T. Kloks, Approximating Treewidth, Pathwidth, Frontsize, and Shortest Elimination Tree, J. Algorithms 18, 2 (1995) 221--237.

....a graph, and focus our efforts on finding parallel elimination orders with linear fill. 1. 4 Related work: Height Ignoring fill, computing an elimination order for a given graph with minimum height is NP hard [33] and remains so even if an additive error in the estimate of the height is allowed [5]. Pan and Reif give one of the first analyses of the parallel height of nested dissection orders and show how nested dissection can be used for solving the shortest path problem in graphs [31, 30] Bodlaender et al. 5] uses an approach similar to [1] to find elimination orders with bounds on the ....

.... remains so even if an additive error in the estimate of the height is allowed [5] Pan and Reif give one of the first analyses of the parallel height of nested dissection orders and show how nested dissection can be used for solving the shortest path problem in graphs [31, 30] Bodlaender et al. [5] uses an approach similar to [1] to find elimination orders with bounds on the height and several related parameters. Both these papers [1, 5] give elimination orders with height at most O(log 2 n) times the minimum possible, for any n node graph. Numerous heuristics without performance ....

[Article contains additional citation context not shown here]

Hans L. Bodlaender, John R. Gilbert, Hj'almt'yr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. Journal of Algorithms, 18(2):238--255, March 1995.

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H. L. Bodlaender, J. R. Gilbert, H. Hafsteinsson, and T. Kloks, Approximating treewidth, pathwidth, frontsize, and shortest elimination tree, Journal of Algorithms, 18 (1995), pp. 238--255.

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H. Bodlaender, R. J. Gilbert, H. Hafsteinson, T. Kloks, Approximating treewidth, pathwidth, frontsize, and shortest elimination tree, J. Algorithms 18 (1995) 238-255.

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BODLAENDER, H. L., GILBERT, J. R., HAFSTEINSSON, H., AND KLOKS, T. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. Journal of Algorithms 18 (1995), 238--255.

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H. Bodlaender, J. Gilbert, H. Hafsteinsson, and T. Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. Journal of Algorithms, 18:238-255, 1995.

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H.L. Bodlaender, J.R. Gilbert, H. Hafsteinsson, and T. Kloks, Approximating treewidth, pathwidth, frontsize, and shortest elimination tree, Journal of Algorithms.

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Hans L. Bodlaender, John R. Gilbert, Hjalmtyr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms, 18(2):238-255, 1995.

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Hans L. Bodlaender, John R. Gilbert, Hjalmtyr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms, 18(2):238--255, 1995.

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Hans L. Bodlaender, John R. Gilbert, Hjalmtyr Hafsteinsson, and Ton Kloks. Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms, 18(2):238--255, 1995.

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H. Bodlaender, J. Gilbert, H. Hafsteinsson, and T. Kloks. Approximating Treewidth, Pathwidth, Frontsize, and Shortest Elimination Tree. Journal of Algorithms, 18:238--155, 1995.

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H. Bodlaender, J. Gilbert, H. Hafsteinsson, and T. Kloks. Approximating Treewidth, Pathwidth, Frontsize, and Shortest Elimination Tree. Journal of Algorithms 18:238-255 (1995). 11

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H. Bodlaender, J. Gilbert, H. Hafsteinsson, and T. Kloks. Approximating Treewidth, Pathwidth, Frontsize, and Shortest Elimination Tree. Journal of Algorithms 18:238-255 (1995).

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