The Complexity of the Cover Polynomials for Planar Graphs of Bounded Degree

  • Markus Bläser
  • Radu Curticapean
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6907)

Abstract

The cover polynomials are bivariate graph polynomials that can be defined as weighted sums over all path-cycle covers of a graph. In [3], a dichotomy result for the cover polynomials was proven, establishing that their evaluation is #P-hard everywhere but at a finite set of points, where evaluation is in FP. In this paper, we show that almost the same dichotomy holds when restricting the evaluation to planar graphs. We even provide hardness results for planar DAGs of bounded degree. For particular subclasses of planar graphs of bounded degree and for variants thereof, we also provide algorithms that allow for polynomial-time evaluation of the cover polynomials at certain new points by utilizing Valiant’s holographic framework.

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References

  1. 1.
    Averbouch, I., Godlin, B., Makowsky, J.A.: A most general edge elimination polynomial. In: Broersma, H., Erlebach, T., Friedetzky, T., Paulusma, D. (eds.) WG 2008. LNCS, vol. 5344, pp. 31–42. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  2. 2.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Computing the Tutte polynomial in vertex-exponential time. In: FOCS 2008, pp. 677–686 (2008)Google Scholar
  3. 3.
    Bläser, M., Dell, H.: Complexity of the cover polynomial. In: ICALP, pp. 801–812 (2007)Google Scholar
  4. 4.
    Bläser, M., Dell, H., Fouz, M.: Complexity and approximability of the cover polynomial. Computation Complexity (accepted)Google Scholar
  5. 5.
    Bläser, M., Dell, H., Makowsky, J.A.: Complexity of the bollobás-riordan polynomial. In: Hirsch, E.A., Razborov, A.A., Semenov, A., Slissenko, A. (eds.) Computer Science – Theory and Applications. LNCS, vol. 5010, pp. 86–98. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  6. 6.
    Bläser, M., Hoffmann, C.: On the complexity of the interlace polynomial. In: Albers, S., Weil, P. (eds.) 25th International Symposium on Theoretical Aspects of Computer Science (STACS 2008), Dagstuhl, Germany, pp. 97–108 (2008)Google Scholar
  7. 7.
    Bollobás, B., Riordan, O.: A Tutte polynomial for coloured graphs. Combinatorics, Probability and Computing 8(1-2), 45–93 (1999)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Cai, J.-y., Lu, P., Xia, M.: Holographic algorithms with matchgates capture precisely tractable planar #CSP. CoRR abs/1008.0683 (2010)Google Scholar
  9. 9.
    Chung, F.R.K., Graham, R.L.: On the cover polynomial of a digraph. J. Combin. Theory Ser. B 65, 273–290 (1995)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Courcelle, B., Makowsky, J.A., Rotics, U.: On the fixed parameter complexity of graph enumeration problems definable in monadic second-order logic. Discrete Applied Mathematics 108(1-2), 23–52 (2001)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    D’Antona, O.M., Munarini, E.: The cycle-path indicator polynomial of a digraph. Advances in Applied Mathematics 25(1), 41–56 (2000)MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    Goldberg, L.A., Jerrum, M.: Inapproximability of the Tutte polynomial. Inform. Comput. 206(7), 908–929 (2008)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Goldberg, L.A., Jerrum, M.: Inapproximability of the Tutte polynomial of a planar graph. CoRR abs/0907.1724 (2009)Google Scholar
  14. 14.
    Hoffmann, C.: A most general edge elimination polynomial - thickening of edges. Fundam. Inform. 98(4), 373–378 (2010)MATHGoogle Scholar
  15. 15.
    Jaeger, F., Vertigan, D.L., Welsh, D.J.A.: On the computational complexity of the Jones and Tutte polynomials. Math. Proc. of the Cambridge Phil. Society 108(1), 35–53 (1990)MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    Linial, N.: Hard enumeration problems in geometry and combinatorics. SIAM J. Algebraic Discrete Methods 7(2), 331–335 (1986)MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    Lotz, M., Makowsky, J.A.: On the algebraic complexity of some families of coloured Tutte polynomials. Advances in Applied Mathematics 32(1), 327–349 (2004)MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    Makowsky, J.A.: From a zoo to a zoology: Towards a general theory of graph polynomials. Theory of Computing Systems 43(3), 542–562 (2008)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Valiant, L.G.: Holographic algorithms (extended abstract). In: FOCS 2004: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science, pp. 306–315. IEEE Computer Society, Washington, DC (2004)CrossRefGoogle Scholar
  20. 20.
    Vertigan, D.L.: The computational complexity of Tutte invariants for planar graphs. SIAM J. Comput. 35(3), 690–712 (2005)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Xia, M., Zhang, P., Zhao, W.: Computational complexity of counting problems on 3-regular planar graphs. Theoretical Computer Science 384(1), 111–125 (2007)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Berlin Heidelberg 2011

Authors and Affiliations

  • Markus Bläser
    • 1
  • Radu Curticapean
    • 1
  1. 1.Department of Computer ScienceSaarland UniversityGermany

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