Combinatorial and geometric approaches to counting problems on linear matroids, graphic arrangements, and partial orders

  • Hiroshi Imai
  • Satoru Iwata
  • Kyoko Sekine
  • Kensyu Yoshida
Session 2
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1090)


For typical #P-hard problems on graphs, we have recently proposed an approach to solve those problems of moderate size rigorously by means of the binary decision diagram, BDD [12, 13]. This paper extends this approach to counting problems on linear matroids, graphic arrangements and partial orders, most of which are already known to be #P-hard, with using geometric properties. Efficient algorithms are provided to the following problems.
  • Computing the BDD representing all bases of a binary or ternary matroid in an output-size sensitive manner; by using this BDD, the Tutte polynomial of the matroid and the weight enumeration of an (n, k) linear code over GF(2) and GF(3) can be computed in time proportional to the size of the BDD.

  • Computing the Tutte polynomial of a linear matroid over the reals via the arrangement construction algorithm in computational geometry.

  • Computing the number of acyclic orientations of a graph, i.e., the number of cells in the corresponding graphic arrangement, and further the number of its lower-dimensional faces.

  • Computing the number of ideals in a partially ordered set, i.e., the number of some faces of the corresponding cone in the graphic arrangement


Span Tree Partial Order Boolean Function Binary Decision Diagram Hasse Diagram 
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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  1. 1.
    M. D. Atkinson and H. W. Chang: Computing the Number of Merging with Constraints. Information Processing Letters, Vol.24 (1987), pp.289–292.Google Scholar
  2. 2.
    D. Avis and K. Fukuda: A Pivoting Algorithm for Convex Hulls and Vertex Enumeration of Arrangements and Polyhedra. Discrete and Computational Geometry, Vol.8 (1992), pp.295–313.Google Scholar
  3. 3.
    G. Brightwell and P. Winkler: Counting Linear Extensions is #P-Complete. Proceedings of the 23rd Annual ACM Symposium on Theory of Computing, 1991, pp.175–181.Google Scholar
  4. 4.
    T. Brylawski and J. Oxley: The Tutte Polynomial and Its Applications. In “Matroid Applications” (N. White, ed.), Encyclopedia of Mathematics and Its Applications, Vol.40 (1992), pp.123–225.Google Scholar
  5. 5.
    M. Dyer, A. Frieze and R. Kannan: A Random Polynomial Time Algorithm for Approximating the Volume of Convex Bodies. Journal of the Association for Computing Machinery, Vol.38 (1991), pp.1–17.Google Scholar
  6. 6.
    H. Edelsbrunner: Algorithms in Combinatorial Geometry. Springer-Verlag, Heidelberg, 1987.Google Scholar
  7. 7.
    C. Greene and T. Zaslavsky: On the Interpretation of Whitney Numbers Through Arrangements of Hyperplanes, Zonotopes, non-Radon partitions and Orientations of Graphs. Transactions of the American Mathematical Society, Vol.280 (1983), pp.97–126.Google Scholar
  8. 8.
    K. Hayase, K. Sadakane and S. Tani: Output-size Sensitiveness of OBDD Construction Through maximal Independent Set Problem. Proceedings of the Conference on Computing and Combinatorics (COCOON'95), Lecture Notes in Computer Science, Vol.959 (1995), pp.229–234.Google Scholar
  9. 9.
    H. Imai: Network-Flow Algorithms for Lower-Truncated Transversal Polymatroids. Journal of the Operations Research Society of Japan, Vol.26, No.3 (1983), pp.186–210.Google Scholar
  10. 10.
    L. Khachiyan: Complexity of Polytope Volume Computation. In “New Trends in Discrete and Computational Geometry” (J. Pach, ed.), Algorithms and Combinatorics, Vol.10, Springer-Verlag, 1993, pp.91–101.Google Scholar
  11. 11.
    J. Oxley: Matroid Theory. Oxford University Press, Oxford, 1992.Google Scholar
  12. 12.
    K. Sekine, H. Imai and S. Tani: Computing the Tutte Polynomial of a Graph of Moderate Size. Proceedings of the 5th International Symposium on Algorithms and Computation (ISAAC'95), Lecture Notes in Computer Science, to appear.Google Scholar
  13. 13.
    K. Sekine and H. Imai: A Unified Approach via BDD to the Network Reliability and Path Numbers. Technical Report, Department of Information Science, University of Tokyo, 1995.Google Scholar
  14. 14.
    P. D. Seymour: Matroid representation over GF(3), Journal of Combinatorial Theory, Ser. B, Vol.26 (1979), pp.159–173.Google Scholar
  15. 15.
    S. Tani and H. Imai: A Reordering Operation for an Ordered Binary Decision Diagram and an Extended Framework for Combinatorics of Graphs. Proceedings of the 5th International Symposium on Algorithms and Computation (ISAAC'94), Lecture Notes in Computer Science, Vol.834 (1994), pp. 575–583.Google Scholar
  16. 16.
    D. J. A. Welsh: Matroid Theory. Academic Press, London, 1976.Google Scholar
  17. 17.
    D. J. A. Welsh: Complexity: Knots, Colourings and Counting. London Mathematical Society Lecture Note Series, Vol.186, Cambridge University Press, 1993.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Hiroshi Imai
    • 1
  • Satoru Iwata
    • 2
  • Kyoko Sekine
    • 1
  • Kensyu Yoshida
    • 1
  1. 1.Department of Information ScienceUniversity of TokyoTokyoJapan
  2. 2.Research Institute of Mathematical SciencesKyoto UniversityKyotoJapan

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