Exponential Time Complexity of the Permanent and the Tutte Polynomial

(Extended Abstract)
  • Holger Dell
  • Thore Husfeldt
  • Martin Wahlén
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6198)


The Exponential Time Hypothesis (ETH) says that deciding the satisfiability of n-variable 3-CNF formulas requires time \(\exp(\Omega(n))\). We relax this hypothesis by introducing its counting version #ETH, namely that every algorithm that counts the satisfying assignments requires time \(\exp(\Omega(n))\). We transfer the sparsification lemma for d-CNF formulas to the counting setting, which makes #ETH robust.

Under this hypothesis, we show lower bounds for well-studied #P-hard problems: Computing the permanent of an n×n matrix with m nonzero entries requires time \(\exp(\Omega(m))\). Restricted to 01-matrices, the bound is \(\exp(\Omega(m/\log m))\). Computing the Tutte polynomial of a multigraph with n vertices and m edges requires time \(\exp(\Omega(n))\) at points (x,y) with (x − 1)(y − 1) ≠ 1 and y ∉ {0,±1}. At points (x,0) with \(x \not \in \{0,\pm 1\}\) it requires time \(\exp(\Omega(n))\), and if x = − 2, − 3,..., it requires time \(\exp(\Omega(m))\). For simple graphs, the bound is \(\exp(\Omega(m/\log^3 m))\).


Perfect Matchings Simple Graph Graph Class Satisfying Assignment Tutte Polynomial 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Agrawal, M.: Determinant versus permanent. In: Proceedings of the 25th International Congress of Mathematicians, ICM, vol. 3, pp. 985–997 (2006)Google Scholar
  2. 2.
    Berkowitz, S.J.: On computing the determinant in small parallel time using a small number of processors. Information Processing Letters 18(3), 147–150 (1984)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Björklund, A., Husfeldt, T.: Exact algorithms for exact satisfiability and number of perfect matchings. Algorithmica 52(2), 226–249 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Computing the Tutte polynomial in vertex-exponential time. In: FOCS, pp. 677–686 (2008)Google Scholar
  5. 5.
    Bläser, M., Dell, H.: Complexity of the cover polynomial. In: Arge, L., Cachin, C., Jurdziński, T., Tarlecki, A. (eds.) ICALP 2007. LNCS, vol. 4596, pp. 801–812. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Brylawski, T.: The Tutte polynomial, Matroid theory and its applications. In: Centro Internazionale Matematico Estivo, pp. 125–275 (1982)Google Scholar
  7. 7.
    Giménez, O., Hliněný, P., Noy, M.: Computing the Tutte polynomial on graphs of bounded clique-width. SIAM J. on Discrete Mathematics 20, 932–946 (2006)zbMATHCrossRefGoogle Scholar
  8. 8.
    Goldberg, L.A., Jerrum, M.: The complexity of ferromagnetic Ising with local fields. Combinatorics, Probability and Computing 16(1), 43–61 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Goldberg, L.A., Jerrum, M.: Inapproximability of the Tutte polynomial. Information and Computation 206(7), 908–929 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? Journal of Computer and System Sciences 63(4), 512–530 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Istrail, S.: Statistical mechanics, three-dimensionality and NP-completeness. I. universality of intractability for the partition function of the Ising model across non-planar lattices. In: STOC, pp. 87–96 (2000)Google Scholar
  12. 12.
    Jaeger, F., Vertigan, D.L., Welsh, D.J.: On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Cambridge 108(1), 35–53 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Jerrum, M., Snir, M.: Some exact complexity results for straight-line computations over semirings. Journal of the ACM 29(3), 874–897 (1982)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Koivisto, M.: Partitioning into sets of bounded cardinality. In: IWPEC, pp. 258–263 (2009)Google Scholar
  15. 15.
    Kutzkov, K.: New upper bound for the #3-SAT problem. Information Processing Letters 105(1), 1–5 (2007)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Lawler, E.L.: A note on the complexity of the chromatic number problem. Information Processing Letters 5, 66–67 (1976)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Papadimitriou, C.M.: Computational Complexity. Addison-Wesley, Reading (1994)zbMATHGoogle Scholar
  18. 18.
    Raz, R.: Multi-linear formulas for permanent and determinant are of super-polynomial size. Journal of the ACM 56(2), 1–17 (2009)CrossRefMathSciNetGoogle Scholar
  19. 19.
    Ryser, H.J.: Combinatorial mathematics. Carus Math. Monographs, vol. 14. Mathematical Association of America (1963)Google Scholar
  20. 20.
    Sekine, K., Imai, H., Tani, S.: Computing the Tutte polynomial of a graph of moderate size. In: Staples, J., Katoh, N., Eades, P., Moffat, A. (eds.) ISAAC 1995. LNCS, vol. 1004, pp. 224–233. Springer, Heidelberg (1995)CrossRefGoogle Scholar
  21. 21.
    Sokal, A.D.: Chromatic roots are dense in the whole complex plane. Combinatorics, Probability and Computing 13(02), 221–261 (2004)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Sokal, A.D.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In: Surveys in Combinatorics. London Mathematical Society Lecture Note Series, vol. 327, pp. 173–226. Cambridge University Press, Cambridge (2005)Google Scholar
  23. 23.
    Valiant, L.G.: The complexity of computing the permanent. Theoretical Computer Science 8(2), 189–201 (1979)zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Holger Dell
    • 1
  • Thore Husfeldt
    • 2
  • Martin Wahlén
    • 3
  1. 1.Humboldt University of BerlinGermany
  2. 2.IT University of Copenhagen, Denmark and Lund UniversitySweden
  3. 3.Lund University, Sweden and Uppsala UniversitySweden

Personalised recommendations