pp 1–16 | Cite as

Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP

  • Cornelius Brand
  • Holger Dell
  • Marc Roth


Jaeger et al. (Math Proc Camb Philos Soc 108(1):35–53, 1990) proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: the evaluation is #P-hard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell, Husfeldt, and Wahlén (in: ICALP 2010, vol. 6198, pp. 426–437, Springer, Berlin, Heidelberg, 2010) and Husfeldt and Taslaman (in: IPEC 2010, vol. 6478, pp. 192–203, Springer, Berlin, Heidelberg, 2010) in combination with the results of Curticapean (in: ICALP 2015, pp. 380–392, Springer, 2015), extended the #P-hardness results to tight lower bounds under the counting exponential time hypothesis #ETH, with the exception of the line \(y=1\), which was left open. We complete the dichotomy theorem for the Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs of a given n-vertex graph cannot be determined in time Open image in new window unless #ETH fails. Another dichotomy theorem we strengthen is the one of Creignou and Hermann (Inf Comput 125(1):1–12, 1996) for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain. We prove that the #P-hard cases cannot be solved in time Open image in new window unless #ETH fails. The main ingredient is to prove that the number of independent sets in bipartite graphs with n vertices cannot be computed in time Open image in new window unless #ETH fails. In order to prove our results, we use the block interpolation idea by Curticapean and transfer it to systems of linear equations that might not directly correspond to interpolation.



We thank Radu Curticapean for various fruitful discussions and interactions, and also Leslie Ann Goldberg, Miki Hermann, Mark Jerrum, John Lapinskas, David Richerby, and all other participants of the “dichotomies” work group at the Simons Institute in the spring of 2016. Moreover, we thank Tyson Williams who in 2013 pointed the second author to [11], and Thore Husfeldt for encouraging us to pursue the publication of the conference version of this article.


  1. 1.
    Björklund, A.: Counting perfect matchings as fast as Ryser. In: Proceedings of the 23rd symposium on discrete algorithms, SODA 2012, pp. 914–921 (2012).
  2. 2.
    Björklund, A., Husfeldt, T., Kaski, P., Koivisto, M.: Computing the Tutte polynomial in vertex-exponential time. In: Proceedings of the 47th annual IEEE symposium on foundations of computer science, FOCS 2008, pp. 677–686 (2008).
  3. 3.
    Cai, J.Y., Chen, X.: Complexity of counting CSP with complex weights. In: Proceedings of the 44th symposium on theory of computing, STOC 2012, pp. 909–920 (2012).
  4. 4.
    Cai, J.Y., Lu, P., Xia, M.: Computational complexity of holant problems. SIAM J. Comput. 40(4), 1101–1132 (2011). MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Calabro, C., Impagliazzo, R., Paturi, R.: A duality between clause width and clause density for SAT. In: Proceedings of the 21st annual IEEE conference on computational complexity, CCC ’06, pp. 252–260. IEEE Computer Society, Washington, DC, USA (2006).
  6. 6.
    Creignou, N., Hermann, M.: Complexity of generalized satisfiability counting problems. Inf. Comput. 125(1), 1–12 (1996). MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Curticapean, R.: Block interpolation: a framework for tight exponential-time counting complexity. In: Proceedings of the 42nd international colloquium on automata, languages and programming, ICALP 2015, pp. 380–392. Springer (2015).
  8. 8.
    Dell, H., Husfeldt, T., Marx, D., Taslaman, N., Wahlén, M.: Exponential time complexity of the permanent and the Tutte polynomial. ACM Trans. Algorithms (2014).
  9. 9.
    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.) International Colloquium on Automata, Languages and Programming, ICALP 2010. Lecture Notes in Computer Science, vol. 6198, pp. 426–437. Springer, Berlin, Heidelberg (2010)Google Scholar
  10. 10.
    Ellis-Monaghan, J., Moffatt, I.: CRC Handbook on the Tutte Polynomial and Related Topics. CRC Press, Boca Raton (2018)Google Scholar
  11. 11.
    Grimmett, G., McDiarmid, C. (eds.): Combinatorics, Complexity, and Chance: A Tribute to Dominic Welsh. Oxford Lecture Series in Mathematics and Its Applications (Book 34). Oxford University Press, Oxford (2007)Google Scholar
  12. 12.
    Husfeldt, T., Taslaman, N.: The exponential time complexity of computing the probability that a graph is connected. In: Raman V., Saurabh S. (eds.) International Symposium on Parameterized and Exact Computation, IPEC 2010. Lecture Notes in Computer Science, vol. 6478, pp. 192–203. Springer, Berlin, Heidelberg (2010)Google Scholar
  13. 13.
    Impagliazzo, R., Paturi, R.: On the complexity of \(k\)-SAT. J. Comput. Syst. Sci. 62(2), 367–375 (2001). MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001). MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Jaeger, F., Vertigan, D.L., Welsh, D.J.: On the computational complexity of the Jones and Tutte polynomials. Math. Proc. Camb. Philos. Soc. 108(1), 35–53 (1990). MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    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). MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Sokal, A.D.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. In: Webb, B.S. (ed.) Surveys in Combinatorics, London Mathematical Society Lecture Note Series, vol. 327, pp. 173–226. Cambridge University Press, Cambridge (2005)Google Scholar
  18. 18.
    Taslaman, N.S.: Exponential-time algorithms and complexity of NP-hard graph problems. Ph.D. thesis, IT-Universitetet i København (2013)Google Scholar
  19. 19.
    Toda, S.: PP is as hard as the polynomial-time hierarchy. SIAM J. Comput. 20(5), 865–877 (1991). MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Valiant, L.G.: The complexity of computing the permanent. Theor. Comput. Sci. 8(2), 189–201 (1979). MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Vertigan, D.L.: On the computational complexity of Tutte, Jones, Homfly and Kauffman invariants. Ph.D. thesis, University of Oxford (1991)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Saarland University and Cluster of Excellence (MMCI)SaarbrückenGermany

Personalised recommendations