The Complexity of Computing the Sign of the Tutte Polynomial (and Consequent #P-hardness of Approximation)

  • Leslie Ann Goldberg
  • Mark Jerrum
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

Abstract

We study the complexity of computing the sign of the Tutte polynomial of a graph. As there are only three possible outcomes (positive, negative, and zero), this seems at first sight more like a decision problem than a counting problem. Surprisingly, however, there are large regions of the parameter space for which computing the sign of the Tutte polynomial is actually #P-hard. As a trivial consequence, approximating the polynomial is also #P-hard in this case. Thus, approximately evaluating the Tutte polynomial in these regions is as hard as exactly counting the satisfying assignments to a CNF Boolean formula. For most other points in the parameter space, we show that computing the sign of the polynomial is in FP, whereas approximating the polynomial can be done in polynomial time with an NP oracle. As a special case, we completely resolve the complexity of computing the sign of the chromatic polynomial — this is easily computable at q = 2 and when q ≤ 32/27, and is NP-hard to compute for all other values of the parameter q.

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References

  1. 1.
    Dell, H., Husfeldt, T., Wahlén, M.: Exponential Time Complexity of the Permanent and the Tutte Polynomial. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 426–437. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  2. 2.
    Goldberg, L.A., Jerrum, M.: Approximating the Partition Function of the Ferromagnetic Potts Model. In: Abramsky, S., Gavoille, C., Kirchner, C., Meyer auf der Heide, F., Spirakis, P.G. (eds.) ICALP 2010. LNCS, vol. 6198, pp. 396–407. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  3. 3.
    Goldberg, L.A., Jerrum, M.: Inapproximability of the Tutte polynomial. Inform. and Comput. 206(7), 908–929 (2008)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Goldberg, L.A., Jerrum, M.: Inapproximability of the Tutte polynomial of a planar graph. CoRR, abs/0907.1724 (2009); To appear ”Computational Complexity” Google Scholar
  5. 5.
    Goldberg, L.A., Jerrum, M.: Approximating the Tutte polynomial of a binary matroid and other related combinatorial polynomials. CoRR, abs/1006.5234 (2010)Google Scholar
  6. 6.
    Jackson, B.: A zero-free interval for chromatic polynomials of graphs. Combinatorics, Probability & Computing 2, 325–336 (1993)MATHCrossRefGoogle Scholar
  7. 7.
    Jackson, B., Sokal, A.D.: Zero-free regions for multivariate Tutte polynomials (alias potts-model partition functions) of graphs and matroids. J. Comb. Theory, Ser. B 99(6), 869–903 (2009)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Jacobsen, J.L., Salas, J.: Is the five-flow conjecture almost false? ArXiv e-prints (September 2010)Google Scholar
  9. 9.
    Jaeger, F., Vertigan, D.L., Welsh, D.J.A.: On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Cambridge Philos. Soc. 108(1), 35–53 (1990)MathSciNetMATHCrossRefGoogle Scholar
  10. 10.
    Provan, J.S., Ball, M.O.: The complexity of counting cuts and of computing the probability that a graph is connected. SIAM J. Comput. 12(4), 777–788 (1983)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Sokal, A.: The multivariate Tutte polynomial. In: Surveys in Combinatorics, Cambridge University Press (2005)Google Scholar
  12. 12.
    Vertigan, D.: The computational complexity of Tutte invariants for planar graphs. SIAM J. Comput. 35(3), 690–712 (electronic) (2005)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Leslie Ann Goldberg
    • 1
  • Mark Jerrum
    • 2
  1. 1.Department of Computer ScienceUniversity of LiverpoolLiverpoolUnited Kingdom
  2. 2.School of Mathematical SciencesQueen Mary, University of LondonLondonUnited Kingdom

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