Advertisement

Generation of Tree Decompositions by Iterated Local Search

  • Nysret Musliu
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4446)

Abstract

Many instances of NP-hard problems can be solved efficiently if the treewidth of their corresponding graph is small. Finding the optimal tree decompositions is an NP-hard problem and different algorithms have been proposed in the literature for generation of tree decompositions of small width. In this paper we propose a novel iterated local search algorithm to find good upper bounds for treewidth of an undirected graph. We propose two heuristics, and their combination for generation of the solutions in the construction phase. The iterated local search algorithm further includes the mechanism for perturbation of solution, and the mechanism for accepting solutions for the next iteration. The proposed algorithm iteratively applies the heuristic for finding good elimination ordering, the acceptance criteria, and the perturbation of solution. We proposed and evaluated different perturbation mechanisms and acceptance criteria. The proposed algorithms are tested on DIMACS instances for vertex coloring, and they are compared with the existing approaches in literature. Our algorithms have a good time performance and for 17 instances improve the best existing upper bounds for the treewidth.

Keywords

Local Search Acceptance Criterion Maximum Clique Local Search Algorithm Construction Phase 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Arnborg, S., Corneil, D.G., Proskurowski, A.: Complexity of finding embeddings in a k-tree. SIAM J. Alg. Disc. Meth. 8, 277–284 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    Bodlaender, H.L.: Discovering treewidth. Technical Report UU-CS-2005-018, Utrecht University (2005)Google Scholar
  3. 3.
    Clautiaux, F., Moukrim, A., Négre, S., Carlier, J.: Heuristic and meta-heurisistic methods for computing graph treewidth. RAIRO Oper. Res. 38, 13–26 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Fulkerson, D.R., Gross, O.A.: Incidence matrices and interval graphs. Pacific Journal of Mathematics 15, 835–855 (1965)zbMATHMathSciNetGoogle Scholar
  5. 5.
    Gavril, F.: Algorithms for minimum coloring, maximum clique, minimum coloring cliques and maximum independent set of a chordal graph. SIAM J. Comput. 1, 180–187 (1972)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gogate, V., Dechter, R.: A complete anytime algorithm for treewidth. In: Proceedings of the 20th Annual Conference on Uncertainty in Artificial Intelligence, UAI-04, pp. 201–208 (2004)Google Scholar
  7. 7.
    Johnson, D.S., Trick, M.A.: The second dimacs implementation challenge: Np-hard problems: Maximum clique, graph coloring, and satisfiability. Series in Discrete Mathematics and Theoretical Computer Science, American Mathematical Society (1993)Google Scholar
  8. 8.
    Kjaerulff, U.: Optimal decomposition of probabilistic networks by simulated annealing. Statistics and Computing 1, 2–17 (1992)Google Scholar
  9. 9.
    Koster, A., Bodlaender, H., van Hoesel, S.: Treewidth: Computational experiments. In: Electronic Notes in Discrete Mathematics 8, Elsevier Science Publishers, Amsterdam (2001)Google Scholar
  10. 10.
    Larranaga, P., Kujipers, C.M.H, Poza, M., Murga, R.H.: Decomposing bayesian networks: triangulation of the moral graph with genetic algorithms. Statistics and Computing (UK) 7(1), 1997 (1991)Google Scholar
  11. 11.
    Robertson, N., Seymour, P.D.: Graph minors. ii. algorithmic aspects of tree-width. Journal Algorithms 7, 309–322 (1986)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Shoikhet, K., Geiger, D.: A practical algorithm for finding optimal triangulations. In: Proc. of National Conference on Artificial Intelligence (AAAI’97), pp. 185–190 (1997)Google Scholar
  13. 13.
    Tarjan, R.E., Yannakakis, M.: Simple linear-time algorithm to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. SIAM J. Comput. 13, 566–579 (1984)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Nysret Musliu
    • 1
  1. 1.Vienna University of Technology, Karlsplatz 13, 1040 ViennaAustria

Personalised recommendations