Computing the Tutte polynomial of a graph of moderate size

  • Kyoko Sekine
  • Hiroshi Imai
  • Seiichiro Tani
Session 7A
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1004)

Abstract

The problem of computing the Tutte polynomial of a graph is #P-hard in general, and any known algorithm takes exponential time at least. This paper presents a new algorithm by exploiting a fact that many 2-isomorphic minors appear in the process of computation. The complexity of the algorithm is analyzed in terms of Bell numbers and Catalan numbers. This algorithm enables us to compute practically the Tutte polynomial of any graph with at most 14 vertices and 91 edges, and that of a planar graph such as 12×12 lattice graph with 144 vertices and 264 edges.

References

  1. 1.
    S. B. Akers: Binary Decision Diagrams. IEEE Trans. on Computers, Vol.C-27(1978), pp.509–516.Google Scholar
  2. 2.
    N. Alon, A. Frieze and D. J. A. Welsh: Polynomial Time Randomised Approximation Schemes for the Tutte Polynomial of Dense Graphs. Proceedings of the IEEE Annual Symposium on Foundations of Computer Science, 1994, pp.24–35.Google Scholar
  3. 3.
    R. E. Bryant: Graph Based Algorithms for Boolean Function Manipulation. IEEE Trans. on Computers, Vol.C-35(1986), pp.677–691.Google Scholar
  4. 4.
    D. R. Karger. A Randomized Fully Polynomial Time Approximation Scheme for the All Terminal Network Reliability Problem. Proceedings of the 27th Annual ACM Symposium on Theory of Computing, 1995, pp.11–17.Google Scholar
  5. 5.
    A. Shioura, A. Tamura and T. Uno: An Optimal Algorithm for Scanning All Spanning Trees of Undirected Graphs. SIAM Journal on Computing, to appear.Google Scholar
  6. 6.
    S. Tani: An Extended Framework of Ordered Binary Decision Diagrams for Combinatorial Graph Problems. Master's Thesis, Department of Information Science, University of Tokyo, 1995.Google Scholar
  7. 7.
    M. B. Thistlethwaite: A Spanning Tree Expansion of the Jones Polynomial. Topology, Vol.26 (1987), pp.297–309.Google Scholar
  8. 8.
    W. T. Tutte: A Contribution to the Theory of Chromatic Polynomials. Canadian Journal of Mathematics, Vol.6 (1954), pp.80–91.Google Scholar
  9. 9.
    D. J. A. Welsh: Complexity: Knots, Colourings and Counting. London Mathematical Society Lecture Note Series, Vol.186, Cambridge University Press, 1993.Google Scholar
  10. 10.
    H. S. Wilf: Algorithms and Complexity. Prentice-Hall, 1986.Google Scholar

Copyright information

© Springer-Verlag 1995

Authors and Affiliations

  • Kyoko Sekine
    • 1
  • Hiroshi Imai
    • 1
  • Seiichiro Tani
    • 2
  1. 1.Department of Information ScienceUniversity of TokyoJapan
  2. 2.Nippon Telegraph and Telephone CorporationJapan

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