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Finding Good Decompositions for Dynamic Programming on Dense Graphs

  • Eivind Magnus Hvidevold
  • Sadia Sharmin
  • Jan Arne Telle
  • Martin Vatshelle
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7112)

Abstract

It is well-known that for graphs with high edge density the tree-width is always high while the clique-width can be low. Boolean-width is a new parameter that is never higher than tree-width or clique-width and can in fact be as small as logarithmic in clique-width. Boolean-width is defined using a decomposition tree by evaluating the number of neighborhoods across the resulting cuts of the graph. Several NP-hard problems can be solved efficiently by dynamic programming when given a decomposition of boolean-width k, e.g. Max Weight Independent Set in time O(n 2 k22k ) and Min Weight Dominating Set in time O(n 2 + nk23k ). Finding decompositions of low boolean-width is therefore of practical interest. There is evidence that computing boolean-width is hard, while the existence of a useful approximation algorithm is still open. In this paper we introduce and study a heuristic algorithm that finds a reasonably good decomposition to be used for dynamic programming based on boolean-width. On a set of graphs of practical relevance, specifically graphs in TreewidthLIB, the best known upper bound on their tree-width is compared to the upper bound on their boolean-width given by our heuristic. For the large majority of the graphs on which we made the tests, the tree-width bound is at least twice as big as the boolean-width bound, and boolean-width compares better the higher the edge density. This means that, for problems like Dominating Set, using boolean-width should outperform dynamic programming by tree-width, at least for graphs of edge density above a certain bound. In view of the amount of previous work on heuristics for tree-width these results indicate that boolean-width could in the future outperform tree-width in practice for a large class of graphs and problems.

Keywords

Local Search Dynamic Programming Edge Density Decomposition Tree Dense Graph 
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.

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References

  1. 1.
    Adler, I., Bui-Xuan, B.M., Rabinovich, Y., Renault, G., Telle, J.A., Vatshelle, M.: On the Boolean-Width of a Graph: Structure and Applications. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 159–170. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  2. 2.
    Belmonte, R., Vatshelle, M.: Graph classes with structured neighborhoods and algorithmic applications. In: Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2011 (2011), www.ii.uib.no/~martinv/Papers/LogBoolw.pdf
  3. 3.
    Bodlaender, H.L.: A linear time algorithm for finding tree-decompositions of small treewidth. SIAM Journal on Computing 25, 1305–1317 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bodlaender, H.L.: Treewidth: Characterizations, Applications, and Computations. In: Fomin, F.V. (ed.) WG 2006. LNCS, vol. 4271, pp. 1–14. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  5. 5.
    Bodlaender, H.L., Koster, A.M.C.A.: Treewidth computations I. Upper bounds. Information and Computation 208, 259–275 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Bodlaender, H.L., Koster, A.M.C.A.: Treewidth computations II. lower bounds. Technical Report UU-CS-2010-022, Department of Information and Computing Sciences, Utrecht University, Utrecht, The Netherlands (2010) (accepted for publication in Information and Computation)Google Scholar
  7. 7.
    Brandstadt, A.: Personal CommunicationGoogle Scholar
  8. 8.
    Bui-Xuan, B.M., Telle, J.A., Vatshelle, M.: Boolean-width of graphs. Theoretical Computer Science (to appear, 2011), www.ii.uib.no/~telle/bib/listofpub/BTV11.pdf
  9. 9.
    Chen, H.: Quantified constraint satisfaction and bounded treewidth. In: de Mántaras, R.L., Saitta, L. (eds.) Proceedings of the 17th European Conference on Artificial Intelligence, ECAI 2004, pp. 161–165 (2004)Google Scholar
  10. 10.
    The second DIMACS implementation challenge: NP-Hard Problems: Maximum Clique, Graph Coloring, and Satisfiability (1992-1993), http://dimacs.rutgers.edu/Challenges/
  11. 11.
    Gottlob, G., Leone, N., Scarcello, F.: A comparison of structural CSP decomposition methods. Acta Informatica 124, 243–282 (2000)MathSciNetzbMATHGoogle Scholar
  12. 12.
    Hicks, I.V., Koster, A.M.C.A., Kolotoğlu, E.: Branch and tree decomposition techniques for discrete optimization. In: Cole Smith, J. (ed.) INFORMS Annual Meeting, TutORials 2005. INFORMS Tutorials in Operations Research Series, ch. 1, pp. 1–29 (2005)Google Scholar
  13. 13.
    Hliněný, P., Oum, S.: Finding branch-decomposition and rank-decomposition. SIAM Journal on Computing 38, 1012–1032 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Kim, K.H.: Boolean matrix theory and its applications. Marcel Dekker (1982)Google Scholar
  15. 15.
    Lauritzen, S.J., Spiegelhalter, D.J.: Local computations with probabilities on graphical structures and their application to expert systems. The Journal of the Royal Statistical Society. Series B (Methodological) 50, 157–224 (1988)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Overwijk, A., Penninkx, E., Bodlaender, H.L.: A Local Search Algorithm for Branchwidth. In: Černá, I., Gyimóthy, T., Hromkovič, J., Jefferey, K., Králović, R., Vukolić, M., Wolf, S. (eds.) SOFSEM 2011. LNCS, vol. 6543, pp. 444–454. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  17. 17.
    Röhrig, H.: Tree decomposition: A feasibility study. Master’s thesis, Max-Planck-Institut für Informatik, Saarbrücken, Germany (1998)Google Scholar
  18. 18.
    Song, Y., Liu, C., Malmberg, R., Pan, F., Cai, L.: Tree decomposition based fast search of RNA structures including pseudoknots in genomes. In: Proceedings of the 2005 IEEE Computational Systems Bioinformatics Conference, CSB 2005, pp. 223–234 (2005)Google Scholar
  19. 19.
  20. 20.
    van Rooij, J.M.M., Bodlaender, H.L., Rossmanith, P.: Dynamic Programming on Tree Decompositions using Generalised Fast Subset Convolution. In: Fiat, A., Sanders, P. (eds.) ESA 2009. LNCS, vol. 5757, pp. 566–577. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  21. 21.
    Zhao, J., Che, D., Cai, L.: Comparative pathway annotation with protein-DNA interaction and operon information via graph tree decomposition. In: Proceedings of Pacific Symposium on Biocomputing, PSB 2007, vol. 12, pp. 496–507 (2007)Google Scholar
  22. 22.
    Zhao, J., Malmberg, R.L., Cai, L.: Rapid ab initio prediction of RNA pseudoknots via graph tree decomposition. Journal of Mathematical Biology 56(1-2), 145–159 (2008)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Eivind Magnus Hvidevold
    • 1
  • Sadia Sharmin
    • 1
  • Jan Arne Telle
    • 1
  • Martin Vatshelle
    • 1
  1. 1.Department of InformaticsUniversity of BergenNorway

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