Exponential Time Complexity of the Permanent and the Tutte Polynomial

(Extended Abstract)
  • Holger Dell
  • Thore Husfeldt
  • Martin Wahlén
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 \(\exp(\Omega(m/\log^3 m))\).


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  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)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Björklund, A., Husfeldt, T.: Exact algorithms for exact satisfiability and number of perfect matchings. Algorithmica 52(2), 226–249 (2008)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefGoogle 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)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Goldberg, L.A., Jerrum, M.: Inapproximability of the Tutte polynomial. Information and Computation 206(7), 908–929 (2008)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Papadimitriou, C.M.: Computational Complexity. Addison-Wesley, Reading (1994)MATHGoogle 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)MATHCrossRefMathSciNetGoogle 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)MATHCrossRefMathSciNetGoogle 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

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