Block Interpolation: A Framework for Tight Exponential-Time Counting Complexity

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9134)


We devise a framework for proving tight lower bounds under the counting exponential-time hypothesis \(\mathsf {\#ETH}\) introduced by Dell et al. Our framework allows to convert many known \(\mathsf {\#P}\)-hardness results for counting problems into results of the following type: If the given problem admits an algorithm with running time \(2^{o(n)}\) on graphs with \(n\) vertices and \(\mathcal {O}(n)\) edges, then \(\mathsf {\#ETH}\) fails. As exemplary applications of this framework, we obtain such tight lower bounds for the evaluation of the zero-one permanent, the matching polynomial, and the Tutte polynomial on all non-easy points except for two lines.


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  1. 1.
    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
  2. 2.
    Bulatov, A.A.: The Complexity of the Counting Constraint Satisfaction Problem. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewicz, I. (eds.) ICALP 2008, Part I. LNCS, vol. 5125, pp. 646–661. Springer, Heidelberg (2008) CrossRefGoogle Scholar
  3. 3.
    Cai, J.-Y., Chen, X.: Complexity of counting CSP with complex weights. STOC 2012, 909–920 (2012)MathSciNetGoogle Scholar
  4. 4.
    Cai, J.-Y., Lu, P., Xia, M.: A Computational Proof of Complexity of Some Restricted Counting Problems. In: Chen, J., Cooper, S.B. (eds.) TAMC 2009. LNCS, vol. 5532, pp. 138–149. Springer, Heidelberg (2009) CrossRefGoogle Scholar
  5. 5.
    Cai, J., Pinyan, L., Xia, M.: Dichotomy for Holant* problems of boolean domain. SODA 2011, 1714–1728 (2011)Google Scholar
  6. 6.
    Dagum, P., Luby, M.: Approximating the permanent of graphs with large factors. Theor. Comput. Sci. 102(2), 283–305 (1992)MATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Dell, H., Husfeldt, T., Marx, D., Taslaman, N., Wahlen, M.: Exponential time complexity of the permanent and the tutte polynomial. ACM Transactions on Algorithms 10(4), 21 (2014)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Leslie Ann Goldberg and Mark Jerrum: The complexity of computing the sign of the tutte polynomial. SIAM J. Comput. 43(6), 1921–1952 (2014)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Hoffmann, C.: Exponential Time Complexity of Weighted Counting of Independent Sets. In: Raman, V., Saurabh, S. (eds.) IPEC 2010. LNCS, vol. 6478, pp. 180–191. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  10. 10.
    Husfeldt, T., Taslaman, N.: The Exponential Time Complexity of Computing the Probability That a Graph Is Connected. In: Raman, V., Saurabh, S. (eds.) IPEC 2010. LNCS, vol. 6478, pp. 192–203. Springer, Heidelberg (2010) CrossRefGoogle Scholar
  11. 11.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Computer and Sys. Sci. 63(4), 512–530 (2001)MATHMathSciNetCrossRefGoogle Scholar
  12. 12.
    Jaeger, F., Vertigan, D.L., Welsh, D J.A.: On the computational complexity of the Jones and Tutte polynomials. Mathematical Proceedings of the Cambridge Philosophical Society 108(1), 35–53 (1990)Google Scholar
  13. 13.
    Jerrum, M., Sinclair, A., Vigoda, E.: A polynomial-time approximation algorithm for the permanent of a matrix with nonnegative entries. J. ACM 51(4), 671–697 (2004)MATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Linial, N.: Hard enumeration problems in geometry and combinatorics. SIAM Journal on Algebraic and Discrete Methods 7(2), 331–335 (1986)MATHMathSciNetCrossRefGoogle Scholar
  15. 15.
    Sokal, A.D.: The multivariate Tutte polynomial (alias Potts model) for graphs and matroids. Surveys in Combinatorics 327, 173–226 (2005)MathSciNetGoogle Scholar
  16. 16.
    Vadhan, S.P.: The complexity of counting in sparse, regular, and planar graphs. SIAM J. Comput. 31(2), 398–427 (2001)MATHMathSciNetCrossRefGoogle Scholar
  17. 17.
    Valiant, L.G.: The complexity of computing the permanent. Theoretical Computer Science 8(2), 189–201 (1979)MATHMathSciNetCrossRefGoogle Scholar

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© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  1. 1.Department of Computer ScienceSaarland UniversitySaarbrückenGermany
  2. 2.Institute for Computer Science and ControlHungarian Academy of Sciences (MTA SZTAKI)BudapestHungary

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