Advertisement

A General Algorithmic Scheme for Modular Decompositions of Hypergraphs and Applications

  • Michel HabibEmail author
  • Fabien de Montgolfier
  • Lalla Mouatadid
  • Mengchuan Zou
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11638)

Abstract

We study here algorithmic aspects of modular decomposition of hypergraphs. In the literature one can find three different definitions of modules, namely: the standard one [19], the k-subset modules [6] and the Courcelle’s one [11]. Using the fundamental tools defined for combinatorial decompositions such as partitive and orthogonal families, we directly derive a linear time algorithm for Courcelle’s decomposition. Then we introduce a general algorithmic tool for partitive families and apply it for the other two definitions of modules to derive polynomial algorithms. For standard modules it leads to an algorithm in \(O(n^3 \cdot l)\) time (where n is the number of vertices and l is the sum of the size of the edges). For k-subset modules we obtain \(O(n^3\cdot m\cdot l)\) (where m is the number of edges). This is an improvement from the best known algorithms for k-subset modular decomposition, which was not polynomial w.r.t. n and m, and is in \(O(n^{3k-5})\) time [6] where k denotes the maximal size of an edge. Finally we focus on applications of orthogonality to modular decompositions of tournaments, simplifying the algorithm from [18]. The question of designing a linear time algorithms for the standard modular decomposition of hypergraphs remains open.

References

  1. 1.
    Berge, C.: Graphes et hypergraphes. Dunod, Paris (1970)zbMATHGoogle Scholar
  2. 2.
    Bergeron, A., Chauve, C., de Montgolfier, F., Raffinot, M.: Computing common intervals of K permutations, with applications to modular decomposition of graphs. SIAM J. Discrete Math. 22(3), 1022–1039 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Billera, L.J.: Clutter decomposition and monotonic boolean functions. Ann. N.-Y. Acad. Sci. 175, 41–48 (1970)MathSciNetzbMATHCrossRefGoogle Scholar
  4. 4.
    Billera, L.J.: On the composition and decomposition of clutters. J. Combin. Theory, Ser. B 11(3), 234–245 (1971)MathSciNetzbMATHCrossRefGoogle Scholar
  5. 5.
    Bioch, J.C.: The complexity of modular decomposition of boolean functions. Discrete Appl. Math. 149(1–3), 1–13 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bonizzoni, P., Vedova, G.D.: An algorithm for the modular decomposition of hypergraphs. J. Algorithms 32(2), 65–86 (1999)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Borassi, M., Crescenzi, P., Habib, M.: Into the square: on the complexity of some quadratic-time solvable problems. Electr. Notes Theor. Comput. Sci. 322, 51–67 (2016)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Capelle, C., Habib, M., de Montgolfier, F.: Graph decompositions and factorizing permutations. Discrete Math. Theor. Comput. Sci. 5(1), 55–70 (2002)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Charbit, P., Habib, M., Limouzy, V., de Montgolfier, F., Raffinot, M., Rao, M.: A note on computing set overlap classes. Inf. Process. Lett. 108(4), 186–191 (2008)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Chein, M., Habib, M., Maurer, M.C.: Partitive hypergraphs. Discrete Math. 37(1), 35–50 (1981)MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    Courcelle, B.: A monadic second-order definition of the structure of convex hypergraphs. Inf. Comput. 178(2), 391–411 (2002)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Dahlhaus, E.: Parallel algorithms for hierarchical clustering and applications to split decomposition and parity graph recognition. J. Algorithms 36(2), 205–240 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Habib, M., McConnell, R.M., Paul, C., Viennot, L.: Lex-BFS and partition refinement, with applications to transitive orientation, interval graph recognition and consecutive ones testing. Theor. Comput. Sci. 234(1–2), 59–84 (2000)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    Habib, M., Paul, C.: A survey of the algorithmic aspects of modular decomposition. Comput. Sci. Rev. 4(1), 41–59 (2010)zbMATHCrossRefGoogle Scholar
  15. 15.
    James, L.O., Stanton, R.G., Cowan, D.D.: Graph decomposition for undirected graphs. In: Proceedings of the 3rd Southeastern International Conference on Combinatorics, Graph Theory, and Computing, pp. 281–290 (1972)Google Scholar
  16. 16.
    McConnell, R.M.: A certifying algorithm for the consecutive-ones property. In: SODA, Proceedings of the Fifteenth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 768–777 (2004)Google Scholar
  17. 17.
    McConnell, R.M., de Montgolfier, F.: Algebraic operations on PQ trees and modular decomposition trees. In: Kratsch, D. (ed.) WG 2005. LNCS, vol. 3787, pp. 421–432. Springer, Heidelberg (2005).  https://doi.org/10.1007/11604686_37CrossRefGoogle Scholar
  18. 18.
    McConnell, R.M., de Montgolfier, F.: Linear-time modular decomposition of directed graphs. Discrete Appl. Math. 145(2), 198–209 (2005)MathSciNetzbMATHCrossRefGoogle Scholar
  19. 19.
    Möhring, R., Radermacher, F.: Substitution decomposition for discrete structures and connections with combinatorial optimization. In: Proceedings of the Workshop on Algebraic Structures in Operations Research, pp. 257–355 (1984)CrossRefGoogle Scholar
  20. 20.
    Möhring, R.H.: Algorithmic aspects of the substitution decomposition in optimization over relations, set systems and boolean functions. Ann. Oper. Res. 4, 195–225 (1985)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Michel Habib
    • 1
    • 3
    Email author
  • Fabien de Montgolfier
    • 1
    • 3
  • Lalla Mouatadid
    • 2
  • Mengchuan Zou
    • 1
    • 3
  1. 1.IRIF, UMR 8243 CNRS & Université de ParisParisFrance
  2. 2.Department of Computer ScienceUniversity of TorontoTorontoCanada
  3. 3.Gang Project, InriaParisFrance

Personalised recommendations