S. Arnborg, J. Lagergren, and D. Seese, Problems easy for tree-decomposable graphs, Journal of Algorithms, 12 (1991), pp. 308--340.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....approach, first introduced by Arnborg and Proskurowski [AP89] Bodlaender [Bod97] gave a better presentation of this technique. Other approaches applicable for solving problems on graphs of bounded treewidth are graph reduction [ACPS93,BdF96] and describing the problems in certain types of logic [ALS88,Cou90] 2.3 Fixed parameter tractability Developing practical algorithms for NP hard problems is an important issue. Recently, Downey and Fellows [DF99] introduced a new approach to cope with this NP hardness, namely fixedparameter tractability. For many NP complete problems, the inherent ....

Stefan Arnborg, Jens Lagergren, and Detlef Seese. Problems easy for tree-decomposable graphs (extended abstract). In Automata, languages and programming (Tampere, 1988), pages 38--51. Springer, Berlin, 1988.

....partially triangulated (j j) grid is any graph obtained by adding noncrossing edges between pairs of nonconsecutive vertices on a common face of a planar embedding of an (j j) grid. 10 For a wide source of parameters satisfying condition (1) we refer to the theory of Courcelle [13] see also [6]) For parameters where f(bw(G) 2 O(bw(G) this result is a strong generalization of Alber et al. s approach which requires that the problem of checking whether p(G) k should satisfy the layerwise separation property [3] Moreover, the algorithms involved are expected to have better ....

S. Arnborg, J. Lagergren, and D. Seese, Problems easy for tree-decomposable graphs (extended abstract), in Automata, languages and programming (Tampere,

....This technique also appeared in a paper written by Bern et al. BLW87] Bodlaender [Bod97] described a better presentation of this technique. Other approaches applicable for solving problems on graphs of bounded treewidth are graph reduction [ACPS93,BdF96] and describing the problems in logic [ALS88,Cou90]. The class of graphs of bounded treewidth is of limited size; we would like to solve NP complete problems for wider classes of graphs. Baker [Bak94] developed several approximation algorithms to solve NPcomplete problems for planar graphs. To extend these algorithms to other graph families, ....

....connectives are and, or and not. Quantifiers 8 and 9 can be applied to the variables. In addition, it has four binary relations: set membership (s 2 S) adjacency test for vertices (adj(u; v) incidency test for vertices and edges (inc(v; e) and equality for variables (a = b) Arnborg et al. [ALS88] extend MSOL to extended monadic second order logic (EMSOL) to have counting or summing evaluations over sets. This feature allows us to define minimization or maximization problems in monadic second order logic. They show that problems definable in EMSOL have linear time or polynomial time ....

[Article contains additional citation context not shown here]

Stefan Arnborg, Jens Lagergren, and Detlef Seese. Problems easy for tree-decomposable graphs (extended abstract). In Automata, languages and programming (Tampere, 1988), pages 38--51. Springer, Berlin, 1988.

....rst introduced by Arnborg and Proskurowski [AP89] Bodlaender [Bod97] described a better presentation of this technique. Other approaches applicable for solving problems on graphs of bounded treewidth are graph reduction [ACPS93,BdF96] and describing the problems in certain types of logic [ALS88,Cou90] 2.3 Fixed parameter tractability Developing practical algorithms for NP hard problems is an important issue. Recently, Downey and Fellows [DF99] introduced a new approach to cope with this NP hardness, namely xed parameter tractability. For many NP complete problems, the inherent ....

Stefan Arnborg, Jens Lagergren, and Detlef Seese. Problems easy for tree-decomposable graphs (extended abstract). In Automata, languages and programming (Tampere, 1988), pages 38-51. Springer, Berlin, 1988.

....approach, first introduced by Arnborg and Proskurowski [6] Bodlaender [15] described a bet ter presentation of this technique. Other approaches applicable for solving problems on graphs of bounded treewidth are graph reduction [4, 17] and describing the problems in certain types of logic [5, 24]. 2.3 Fixed parameter tractability Developing practical algorithms for NP hard problems is an important issue. Recently, Downey and Fellows [29] introduced a new approach to cope with this NP hardness, namely fixed parameter tractability. For many NP complete problems, the inherent combinatorial ....

Stefan Arnborg, Jens Lagergren, and Detlef Seese. Problems easy for tree-decomposable graphs (extended abstract). In Automata, languages and programming (Tampere, 1988), pages 38 51. Springer, Berlin, 1988.

.... treewidth, also known as partial k trees [21] Graphs of bounded treewidth are known for their good algorithmic properties; in particular, any problem that can be expressed in the language of extended monadic second order logic (EMSOL) can be solved using a general dynamic programming approach [1]. Although the subgraph isomorphism problem can be expressed in EMSOL when the source graph is xed, additional restrictions are needed to solve the two input problem in polynomial time. Matou sek and Thomas [18] presented an O(n ) time algorithm for bounded degree partial k trees, and an O(n ....

Stefan Arnborg, Jens Lagergren, and Detlef Seese. Problems easy for tree-decomposable graphs (extended abstract). In Proc., 15th Internat. Colloquium on Automata, Languages, and Programming, pages 38-51, 1998.

....first introduced by Arnborg and Proskurowski [AP89] Bodlaender [Bod97] described a better presentation of this technique. Other approaches applicable for solving problems on graphs of bounded treewidth are graph reduction [ACPS93,BdF96] and describing the problems in certain types of logic [ALS88,Cou90] 2.3 Fixed parameter tractability Developing practical algorithms for NP hard problems is an important issue. Recently, Downey and Fellows [DF99] introduced a new approach to cope with this NP hardness, namely fixed parameter tractability. For many NP complete problems, the inherent ....

Stefan Arnborg, Jens Lagergren, and Detlef Seese. Problems easy for tree-decomposable graphs (extended abstract). In Automata, languages and programming (Tampere, 1988), pages 38--51. Springer,

....first introduced by Arnborg and Proskurowski [AP89] Bodlaender [Bod97] described a better presentation of this technique. Other approaches applicable for solving problems on graphs of bounded treewidth are graph reduction [ACPS93,BdF96] and describing the problems in certain types of logic [ALS88,Cou90] 2.3 Fixed parameter tractability Developing practical algorithms for NP hard problems is an important issue. Recently, Downey and Fellows [DF99] introduced a new approach to cope with this NP hardness, namely fixed parameter tractability. For many NP complete problems, the inherent ....

Stefan Arnborg, Jens Lagergren, and Detlef Seese. Problems easy for tree-decomposable graphs (extended abstract). In Automata, languages and programming (Tampere, 1988), pages 38--51. Springer,

....computers and intractability [GJ79] 2.2 Treewidth Graphs of bounded treewidth, described in this section, are known for their good algorithmic properties. Many problems which are intractable in the general case can be solved in polynomial time or even linear time on graphs of bounded treewidth [ALS88] The notion of treewidth was introduced by Robertson and Seymour [RS86] and plays an important role in their fundamental work on graph minors. To define this notion, first we consider the representation of a graph as a tree, which is the basis of our algorithms. 10 Algorithms for Graphs of ....

....or and not. Quantifiers # and # can be applied to Background 19 the variables. In addition, it has four binary relations: set membership (s # S) adjacency test for vertices (adj(u, v) incidency test for vertices and edges (inc(v, e) and equality for variables (a = b) Arnborg et al. ALS88] extend MSOL to extended monadic second order logic (EMSOL) to have counting or summing evaluations over sets. This feature allows us to define minimization or maximization problems in monadic second order logic. They show that problems definable in EMSOL have linear time or polynomial time ....

[Article contains additional citation context not shown here]

Stefan Arnborg, Jens Lagergren, and Detlef Seese. Problems easy for treedecomposable graphs (extended abstract). In Automata, languages and programming (Tampere,

....Figure 15) 39 8 Tree Width Tree width is a concept in graph theory meant to model how close a given graph is to a tree. More importantly, a bounded tree width specifies a large class of graphs for which polynomial time (and often linear time) algorithms exist for a variety of problems in NP [4,5,13,64,96]. If we can find a tree partition scheme for a given graph, however, it is possible to adapt the tree partition scheme into a tree decomposition, and the tree width is then necessarily bounded. First, however, we must define tree width: Definition 50 Let T = N, L) be a tree with nodes N and ....

....makes our method especially useful for dynamically changing structures, and we are currently investigating the limits of this approach. Graphs with bounded tree width have been recognized as constituting a class of graphs about which many di#cult problems, some in NP, can be solved e# ciently [4,5,13,68,69,71,90,96]. We have taken the opposite approach, starting with a di#cult problem and showing that a certain e#cient solution implies an upper bound on tree width related to the partitionability of the graph. Nevertheless, it is interesting to find our solution converging to the same class of graphs. ....

Stefan Arnborg, Jens Lagergren, and Detlef Seese. Problems easy for tree-decomposable graphs. In Timo Lepisto and Arto Salomaa, editors, Proceedings of the 15th International Colloquium On Automata, Languages and Programming, number 317 in Lecture Notes in Computer Science, pages 38--51, Tampere, Finland, July 11--15, 1988. Springer-Verlag. Extended abstract.

....introduced by Robertson and Seymour [22, 21] seems to often capture a property of the input graph which makes hard problems easy. Various NP hard problems, like Clique or Coloring, have a polynomial time algorithm (linear time in fact) if the input graph has bounded tree width (see for example [2]) We will present the formal definition of tree width in Section 2. Bounded tree width can also be used to obtain good approximation algorithms for those problems that remain NP hard even if restricted to graphs of bounded tree width. In our case, Unrestricted Vertex Multicut is NP hard in ....

S. Arnborg and J. Lagergren, "Problems Easy for Tree-Decomposable Graphs," Journal of Algorithms, 12 (2), 308-340, 1991.

....of width at most k for a graph of treewidth at most k, using the algorithm of [11] to compute a decomposition of width at most k from a decomposition of width at most 2k. Several authors give improved algorithms for graphs of treewidth at most two [9] and treewidth at most four [5] In [6, 4, 7], linear time algorithms for problems on bounded treewidth graphs are given, which are NP hard for general graphs. These algorithms assume that the tree decomposition is given as part of the input. Several authors deal with the problem of computing tree decompositions of bounded treewidth graphs ....

....a tree, we obtain the following corollary. Corollary 2 Given a directed graph G= V;E) with treewidth at most k, it takes O(sort(N) I Os to compute a shortest path tree for any source vertex s in G. 6. 2 Solving NP hard Problems on Graphs of Bounded Treewidth Arnborg, Lagergren, and Seese [6] present polynomial time solutions for NP hard problems on bounded treewidth graphs which are describable in monadic second order logic. Arnborg and Proskurowski [4] present linear time algorithms for graphs of bounded treewidth by processing a k tree embedding of the graph. It is straightforward ....

Stefan Arnborg, Jens Lagergren, and Detlef Seese. Problems easy for tree-decomposable graphs. In Proceedings of the 15th International Colloquium on Automata, Languages and Programming, volume 317 of Lecture Notes in Computer Science, pages 38--51. Springer Verlag, 1988.

.... from efficiently finding close to optimal tree decompositions [Dechter and Pearl, 1989, Amir and McIlraith, 2000] The solution time of many graph related NP hard problems is possible in polynomial time if the graph has low treewidth and a triangulation of minimum treewidth is given (e.g. [Arnborg et al. 1991]) In this paper we present four approximation algorithms for finding triangulations of minimum treewidth. The first algorithm improves an algorithm of [Robertson and Seymour, 1995] and produces factor 4 approximations in time O(2 4.38k n 2 k) where n, k are the number of nodes and the ....

Arnborg, S., Lagergren, J., and Seese, D. (1991). Problems easy for tree-decomposable graphs. J. of

.... the smallest width k of any graph G 0 obtained by triangulation methods from the primal graph G of a CSP such that G 0 ensures k 1 consistency. Graphs having induced width k can be also characterized as partial k trees [ Freuder, 1990 ] or, equivalently, as graphs having treewidth k [ Arnborg et al. 1991 ] It follows that, for fixed k, checking whether w k is feasible in linear time [ Bodlaender, 1997 ] If w is bounded by a constant, a CSP is solvable in polynomial time. The approach to CSPs based on w is referred to as the w Tractability method [ Dechter, 1992 ] Note that this method ....

S. Arnborg, J. Lagergren, and D. Seese. Problems easy for tree-decomposable graphs. J. of Algorithms, 12:308--340, 1991.

....Science, Mathematical Institute, Heinrich Heine University D usseldorf, D40225 D usseldorf, Germany, e mail: wanke cs.uni duesseldorf.de 1 Several authors discussed the problem how to solve NP complete graph problems on tree structured graphs in polynomial time. Three of these approaches are [ALS91, BLW87, Bod88]. Arnborg, Lagergren, and Seese in [ALS91] are interested in giving a very general and logical criterion for a graph property to be decidable in polynomial time when restricted to partial k trees. Bodlaender introduces in [Bod88] two classes of graph problems called ECC and LCC problems that can ....

....D usseldorf, D40225 D usseldorf, Germany, e mail: wanke cs.uni duesseldorf.de 1 Several authors discussed the problem how to solve NP complete graph problems on tree structured graphs in polynomial time. Three of these approaches are [ALS91, BLW87, Bod88] Arnborg, Lagergren, and Seese in [ALS91] are interested in giving a very general and logical criterion for a graph property to be decidable in polynomial time when restricted to partial k trees. Bodlaender introduces in [Bod88] two classes of graph problems called ECC and LCC problems that can be solved in polynomial time on partial ....

[Article contains additional citation context not shown here]

S. Arnborg, J. Lagergren, and D. Seese. Problems easy for tree-decomposable graphs. Journal of Algorithms, 12:308-340, 1991.

.... [3] Computing proximity of web pages on the WWW is done efficiently using a tree decomposition of low treewidth [20] The solution time of many graph related NP hard problems is possible in polynomial time if the graph has low treewidth and a triangulation of minimum treewidth is given (e.g. [5]) Also, there is evidence that many real life instances of the problems we wish to solve have low treewidth [15, 25] 1 Optimally solving TREEWIDTH is known to be NP hard [4] It is an open question whether a constant factor approximation can be found in polynomial time. Nevertheless, several ....

S. Arnborg, J. Lagergren, and D. Seese. Problems easy for tree-decomposable graphs. J. Algorithms, 12:308--340, 1991.

....has to code the treedecomposition of the graphs into trees using standard and folklore ideas of interpretability. The interpretation can in fact be done faithfully. The proof is left to the reader. A similar result is proved in details for a slightly extended notion of interpretability in [1] (see Theorem 4.5) Theorem 7. If K is a class of planar graphs such that Th2(K) or Th,2(K) are decidable. Then there is an n such that each G K has tree width n. Corollary. If K is a class of planar graphs such that Th2(K) or Th,2(K) are decidable, then there is a class T of trees such that ....

....is an n such that each G K has tree width n. Corollary. If K is a class of extended graphs such that Th2(K) or Thw2(K) are decidable, then there is a class T of trees such that Th2(K) is interpretable in Th2(r) First it is easy to observe that Fact 2 holds also for extended graphs (compare [1]) Using this one gets the Corollary from Theorem 8. Theorem 9. Let K be a class of extended graphs such that for each planar graphs H there is an extended graph G K such that H is isomorphic to a minor of G. Then th2(K ) and Th,2(K) are undecidable. This result is an easy consequence of Theorem ....

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S. Arnborg, J. Lagergren and D. Seese, Problems easy for tree-decomposable graphs, extended abstract, in: T. Lepisto and A. Salomaa, eds., Proc. 15th Int. Colloq. Automata, Languages and Programming (Tampere, Finland, 1988.

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S. Arnborg, J. Lagergren, and D. Seese, Problems easy for tree-decomposable graphs, Journal of Algorithms, 12 (1991), pp. 308--340.

No context found.

S. Arnborg, J. Lagergren, and D. Seese, Problems easy for tree-decomposable graphs (extended abstract), in Automata, languages and programming (Tampere,

No context found.

Stefan Arnborg, Jens Lagergren, and Detlef Seese. Problems easy for tree-decomposable graphs (extended abstract). In Automata, languages and programming (Tampere, 1988), pages 38--51. Springer, Berlin, 1988.

No context found.

Stefan Arnborg, Jens Lagergren, and Detlef Seese. Problems easy for tree-decomposable graphs (extended abstract) . In Proc., 15th Internat. Colloquium on Automata, Languages, and Programming, pages 38-51, 1998.

No context found.

S. Arnborg, J. Lagergren, and D. Seese. Problems easy for tree-decomposable graphs. Journal of Algorithms, 12:308--340, 1991.

No context found.

S. Arnborg, J. Lagergren, and D. Seese, Problems easy for tree-decomposable graphs (extended abstract), in Automata, languages and programming (Tampere,

No context found.

S. Arnborg and J. Lagergren, "Problems Easy for Tree-Decomposable Graphs," Journal of Algorithms, 12 (2), 308-340, 1991.

No context found.

S. Arnborg, J. Lagergren, and D. Seese. Problems easy for tree-decomposable graphs (extended abstract). In Proc. I5'th ICALP, pages 38-51, Springer Ver- lag, Lect. Notes in Comp. Sc. 317, 1988.

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