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Combinatorica

, Volume 15, Issue 1, pp 1–10 | Cite as

The complexity of computing the tutte polynomial on transversal matroids

  • Charles J. Colbourn
  • J. Scott Provan
  • Dirk Vertigan
Article

Abstract

The complexity of computing the Tutte polynomialT(M,x,y) is determined for transversal matroidM and algebraic numbersx andy. It is shown that for fixedx andy the problem of computingT(M,x,y) forM a transversal matroid is #P-complete unless the numbersx andy satisfy (x−1)(y−1)=1, in which case it is polynomial-time computable. In particular, the problem of counting bases in a transversal matroid, and of counting various types of “matchable” sets of nodes in a bipartite graph, is #P-complete.

Mathematics Subject Classification (1991)

05 D 15 68 Q 25 68 R 05 

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References

  1. [1]
    M. O. Ball, andJ. S. Provan: Bounds on the reliability polynomial of shellable independence systems,SIAM J. Alg. Disc. Math. 3 (1982), 166–181.zbMATHMathSciNetGoogle Scholar
  2. [2]
    R. A. Brualdi, andE. B. Scrimger: Exchange systems, matchings, and transversals,J. Comb. Th. 5 (1968), 224–257.MathSciNetGoogle Scholar
  3. [3]
    T. Brylawski: Constructions, inTheory of Matroids, N. White ed., Cambridge University Press, New York (1986), 127–223.Google Scholar
  4. [4]
    C. J. Colbourn, andE. S. Elmallah: Reliable assignments of processors to tasks and factoring on matroids,Discrete Math. 114 (1993), 115–129.zbMATHMathSciNetCrossRefGoogle Scholar
  5. [5]
    M. R. Garey, andD. S. Johnson:Computers and Intractibility: A Guide to the Theory of NP-completeness, W. H. Freeman, San Francisco, CA, 1979.zbMATHGoogle Scholar
  6. [6]
    J. J. Harms, andC. J. Colbourn: Probabilistic single processor scheduling.Disc. Appl. Math. 27 (1990), 101–112.zbMATHMathSciNetCrossRefGoogle Scholar
  7. [7]
    F. Jaeger, D. Vertigan, andD. J. A. Welsh: On the computational complexity of the Jones and Tutte polynomials,Math. Proc. Camb. Phil. Soc. 108 (1990) 35–53.zbMATHMathSciNetCrossRefGoogle Scholar
  8. [8]
    G. Kirchoff: Über die Auflösung der Gleichungen auf welche man bei der Untersuchung der Linearen Verteilung Galvansicher Ströme geführt wird,Ann. Phys. Chem. 72 (1847), 497–508.Google Scholar
  9. [9]
    L. G. Valiant: The complexity of enumeration and reliability problems,SIAM J. Computing 8 (1979), 410–421.zbMATHMathSciNetCrossRefGoogle Scholar
  10. [10]
    D. Vertigan:The computations complexity of Tutte invariants for planar graphs, preprint, Mathematical Institute, Oxford University.Google Scholar
  11. [11]
    D. Vertigan:Bicycle dimension and special points of the Tutte polynomial, preprint, Mathematical Institute, Oxford University.Google Scholar
  12. [12]
    D. Vertigan:Counting bases is #P-complete for various classes of matroids, in preparation.Google Scholar

Copyright information

© Akadémiai Kiadó 1995

Authors and Affiliations

  • Charles J. Colbourn
    • 1
  • J. Scott Provan
    • 2
  • Dirk Vertigan
    • 3
  1. 1.Department of Combinatorics and OptimizationUniversity of WaterlooWaterlooCanada
  2. 2.Department of Operations ResearchUniversity of North CarolinaChapel Hill
  3. 3.Department of MathematicsLouisiana State UniversityBaton Rouge

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