A Generic Approach to Decomposition Algorithms, with an Application to Digraph Decomposition

  • Binh-Minh Bui-Xuan
  • Pinar Heggernes
  • Daniel Meister
  • Andrzej Proskurowski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6842)


A set family is a collection of sets over a universe. If a set family satisfies certain closure properties then it admits an efficient representation of its members by labeled trees. The size of the tree is proportional to the size of the universe, whereas the number of set family members can be exponential. Computing such efficient representations is an important task in algorithm design. Set families are usually not given explicitly (by listing their members) but represented implicitly.

We consider the problem of efficiently computing tree representations of set families. Assuming the existence of efficient algorithms for solving the Membership and Separation problems, we prove that if a set family satisfies weak closure properties then there exists an efficient algorithm for computing a tree representation of the set family. The running time of the algorithm will mainly depend on the running times of the algorithms for the two basic problems. Our algorithm generalizes several previous results and provides a unified approach to the computation for a large class of decompositions of graphs. We also introduce a decomposition notion for directed graphs which has no undirected analogue. We show that the results of the first part of the paper are applicable to this new decomposition. Finally, we give efficient algorithms for the two basic problems and obtain an \({\cal O}(n^3)\)-time algorithm for computing a tree representation.


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  1. 1.
    Bernáth, A.: A note on the directed source location algorithm. Technical report, TR-2004-12, Egerváry Research Group, Budapest (2004)Google Scholar
  2. 2.
    Bui-Xuan, B.-M.: Tree-representation of set families in graph decompositions and efficient algorithms. PhD thesis, University of Montpellier II (2008)Google Scholar
  3. 3.
    Bui-Xuan, B.-M., Habib, M.: A representation theorem for union-difference families and application. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 492–503. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  4. 4.
    Bui-Xuan, B.-M., Habib, M., Limouzy, V., de Montgolfier, F.: Algorithmic Aspects of a General Modular Decomposition Theory. Discrete Applied Mathematics 157, 1993–2009 (2009)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Bui-Xuan, B.-M., Habib, M., Rao, M.: Tree-representation of set families and applications to combinatorial decompositions. European Journal of Combinatorics (to appear)Google Scholar
  6. 6.
    Chein, M., Habib, M., Maurer, M.-C.: Partitive hypergraphs. Discrete Mathematics 37, 35–50 (1981)MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    Cunningham, W.: A combinatorial decomposition theory. PhD thesis, University of Waterloo (1973)Google Scholar
  8. 8.
    Cunningham, W., Edmonds, J.: A combinatorial decomposition theory. Canadian Journal of Mathematics 32, 734–765 (1980)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Cunningham, W.: Decomposition of directed graphs. SIAM Journal on Algebraic and Discrete Methods 2, 214–228 (1982)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dinitz, E., Karzanov, A., Lomonosov, M.: On the structure of a family of minimal weighted cuts in a graph. In: Pridman, A. (ed.) Studies in Discrete Optimization, Nauka, Moscow, pp. 290–306 (1976)Google Scholar
  11. 11.
    Edmonds, J., Giles, R.: A min-max relation for submodular functions on graphs. Annals of Discrete Mathematics 1, 185–204 (1977)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gabow, H.: Centroids, Representations, and Submoduar Flows. Journal of Algorithms 18, 586–628 (1995)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Hsu, W.-L., Gabor, C., Supowit, K.: Recognizing circle graphs in polynomial time. Journal of the ACM 36, 435–473 (1989)MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    de Mongolfier, F., Rao, M.: The bi-join decomposition. Electronic Notes in Discrete Mathematics 22, 173–177 (2005)CrossRefGoogle Scholar
  15. 15.
    Queyranne, M.: Minimizing symmetric submodular functions. Mathematical Programming 82, 3–12 (1998)MathSciNetMATHGoogle Scholar
  16. 16.
    Schrijver, A.: Combinatorial Optimization – Polyhedra and Efficiency. Springer, Heidelberg (2003)MATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Binh-Minh Bui-Xuan
    • 1
  • Pinar Heggernes
    • 1
  • Daniel Meister
    • 2
  • Andrzej Proskurowski
    • 3
  1. 1.Department of InformaticsUniversity of BergenNorway
  2. 2.Theoretical Computer ScienceUniversity of TrierGermany
  3. 3.Department of Information and Computer ScienceUniversity of OregonUSA

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