Advertisement

On the Boolean-Width of a Graph: Structure and Applications

  • Isolde Adler
  • Binh-Minh Bui-Xuan
  • Yuri Rabinovich
  • Gabriel Renault
  • Jan Arne Telle
  • Martin Vatshelle
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6410)

Abstract

Boolean-width is a recently introduced graph invariant. Similar to tree-width, it measures the structural complexity of graphs. Given any graph G and a decomposition of G of boolean-width k, we give algorithms solving a large class of vertex subset and vertex partitioning problems in time \(O^*(2^{O(k^2)})\). We relate the boolean-width of a graph to its branch-width and to the boolean-width of its incidence graph. For this we use a constructive proof method that also allows much simpler proofs of similar results on rank-width in [S. Oum. Rank-width is less than or equal to branch-width. Journal of Graph Theory 57(3):239–244, 2008]. For an n-vertex random graph, with a uniform edge distribution, we show that almost surely its boolean-width is Θ(log2 n) – setting boolean-width apart from other graph invariants – and it is easy to find a decomposition witnessing this. Combining our results gives algorithms that on input a random graph on n vertices will solve a large class of vertex subset and vertex partitioning problems in quasi-polynomial time \(O^*(2^{O(\log ^4 n)})\).

Keywords

Random Graph Decomposition Tree Graph Class Boolean Matrix Incidence 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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bodlaender, H., Koster, A.: Treewidth Computations I Upper Bounds. Technical Report UU-CS-2008-032, Department of Information and Computing Sciences, Utrecht University (2008)Google Scholar
  2. 2.
    Bui-Xuan, B.-M., Telle, J.A., Vatshelle, M.: Boolean-width of graphs. In: 4th International Workshop on Parameterized and Exact Computation (IWPEC 2009). LNCS, vol. 5917, pp. 61–74. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  3. 3.
    Bui-Xuan, B.-M., Telle, J.A., Vatshelle, M.: H-join decomposable graphs and algorithms with runtime single exponential in rankwidth. Discrete Applied Mathematics 158(7), 809–819 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Corneil, D., Rotics, U.: On the relationship between clique-width and treewidth. SIAM Journal on Computing 34(4), 825–847 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Courcelle, B.: Graph rewriting: An algebraic and logic approach. In: Handbook of Theoretical Computer Science, Volume B: Formal Models and Sematics (B), pp. 193–242 (1990)Google Scholar
  6. 6.
    Dorn, F.: Dynamic programming and fast matrix multiplication. In: Azar, Y., Erlebach, T. (eds.) ESA 2006. LNCS, vol. 4168, pp. 280–291. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  7. 7.
    Downey, R., Fellows, M.: Parameterized Complexity. Springer, Heidelberg (1999)CrossRefzbMATHGoogle Scholar
  8. 8.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Heidelberg (2006)zbMATHGoogle Scholar
  9. 9.
    Ganian, R., Hliněný, P.: On Parse Trees and Myhill-Nerode-type Tools for handling Graphs of Bounded Rank-width. Discrete Applied Mathematics 158(7), 851–867 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Gerber, M., Kobler, D.: Algorithms for vertex-partitioning problems on graphs with fixed clique-width. Theoretical Computer Science 299(1-3), 719–734 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Hliněný, P., Oum, S., Seese, D., Gottlob, G.: Width parameters beyond tree-width and their applications. The Computer Journal 51(3), 326–362 (2008)CrossRefGoogle Scholar
  12. 12.
    Hvidevold, E.: Implementation of heuristics for computing boolean-width, Master thesis, University of Bergen (September 2010) (to appear)Google Scholar
  13. 13.
    Johansson, Ö.: Clique-decomposition, NLC-decomposition and modular decomposition – Relatiohships and results for random graphs. Congressus Numerantium 132, 39–60 (1998)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Kanté, M.: Vertex-minor reductions can simulate edge contractions. Discrete Applied Mathematics 155(17), 2328–2340 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kim, K.H.: Boolean matrix theory and applications. Marcel Dekker, New York (1982)zbMATHGoogle Scholar
  16. 16.
    Kloks, T., Bodlaender, H.: Only few graphs have bounded treewidth. Technical Report UU-CS-92-35, Department of Information and Computing Sciences, Utrecht University (1992)Google Scholar
  17. 17.
    Kobler, D., Rotics, U.: Edge dominating set and colorings on graphs with fixed clique-width. Discrete Applied Mathematics 126(2-3), 197–221 (2003); Abstract at SODA 2001MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lee, C., Lee, J., Oum, S.: Rank-width of Random Graphs, http://arxiv.org/pdf/1001.0461
  19. 19.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press, Cambridge (1995)CrossRefzbMATHGoogle Scholar
  20. 20.
    Oum, S.: Rank-width is less than or equal to branch-width. Journal of Graph Theory 57(3), 239–244 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Rabinovich, Y., Telle, J.A.: On the boolean-width of a graph: structure and applications, http://arxiv.org/pdf/0908.2765
  22. 22.
    Rooij, J., Bodlaender, H., 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
  23. 23.
    Telle, J.A., Proskurowski, A.: Algorithms for vertex partitioning problems on partial k-trees. SIAM Journal on Discrete Mathematics 10(4), 529–550 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Isolde Adler
    • 1
  • Binh-Minh Bui-Xuan
    • 1
  • Yuri Rabinovich
    • 2
  • Gabriel Renault
    • 1
  • Jan Arne Telle
    • 1
  • Martin Vatshelle
    • 1
  1. 1.Department of InformaticsUniversity of BergenNorway
  2. 2.Department of Computer ScienceHaifa UniversityIsrael

Personalised recommendations