Complexity of the Cover Polynomial

  • Markus Bläser
  • Holger Dell
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4596)

Abstract

The cover polynomial introduced by Chung and Graham is a two-variate graph polynomial for directed graphs. It counts the (weighted) number of ways to cover a graph with disjoint directed cycles and paths, it is an interpolation between determinant and permanent, and it is believed to be a directed analogue of the Tutte polynomial. Jaeger, Vertigan, and Welsh showed that the Tutte polynomial is \(\sharp\)-hard to evaluate at all but a few special points and curves. It turns out that the same holds for the cover polynomial: We prove that, in almost the whole plane, the problem of evaluating the cover polynomial is \(\sharp\)-hard under polynomial-time Turing reductions, while only three points are easy. Our construction uses a gadget which is easier to analyze and more general than the XOR-gadget used by Valiant in his proof that the permanent is \(\sharp\)-complete.

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References

  1. 1.
    Chung, F.R., Graham, R.L.: On the cover polynomial of a digraph. Journal of Combinatorial Theory Series B 65, 273–290 (1995)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Jaeger, F., Vertigan, D.L., Welsh, D.J.: On the computational complexity of the Jones and Tutte polynomials. Mathematical Proceedings of the Cambridge Philosophical Society 108, 35–53 (1990)MATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Lotz, M., Makowsky, J.A.: On the algebraic complexity of some families of coloured Tutte polynomials. Advances in Applied Mathematics 32, 327–349 (2004)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bollobás, B., Riordan, O.: A Tutte polynomial for coloured graphs. Combinatorics, Probability and Computing 8, 45–93 (1999)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Giménez, O., Noy, M.: On the complexity of computing the Tutte polynomial of bicircular matroids. Combinatorics, Probability and Computing 15, 385–395 (2006)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Goldberg, L.A., Jerrum, M.: Inapproximability of the Tutte polynomial (2006)Google Scholar
  7. 7.
    Bulatov, A., Grohe, M.: The complexity of partition functions. Theoretical Computer Science 348, 148–186 (2005)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Dyer, M.E., Goldberg, L.A., Paterson, M.: On counting homomorphisms to directed acyclic graphs. In: Bugliesi, M., Preneel, B., Sassone, V., Wegener, I. (eds.) ICALP 2006. LNCS, vol. 4051, pp. 38–49. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  9. 9.
    Valiant, L.G.: The complexity of computing the permanent. Theoretical Computer Science 8, 189–201 (1979)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    D’Antona, O.M., Munarini, E.: The cycle-path indicator polynomial of a digraph. Adv. Appl. Math. 25, 41–56 (2000)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Bläser, M., Dell, H.: Complexity of the cover polynomial (journal version) (to appear)Google Scholar
  12. 12.
    Dyer, M.E., Frieze, A.M., Jerrum, M.: Approximately counting Hamilton paths and cycles in dense graphs. SIAM Journal on Computing 27, 1262–1272 (1998)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2007

Authors and Affiliations

  • Markus Bläser
    • 1
  • Holger Dell
    • 1
  1. 1.Computational Complexity Group, Saarland UniversityGermany

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