The Complexity of Satisfiability of Small Depth Circuits

  • Chris Calabro
  • Russell Impagliazzo
  • Ramamohan Paturi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5917)


Say that an algorithm solving a Boolean satisfiability problem x on n variables is improved if it takes time poly(|x|)2 cn for some constant c < 1, i.e., if it is exponentially better than a brute force search. We show an improved randomized algorithm for the satisfiability problem for circuits of constant depth d and a linear number of gates cn: for each d and c, the running time is 2(1 − δ)n where the improvement \(\delta\geq 1/O(c^{2^{d-2}-1}\lg^{3\cdot 2^{d-2}-2}c)\), and the constant in the big-Oh depends only on d. The algorithm can be adjusted for use with Grover’s algorithm to achieve a run time of \(2^{\frac{1-\delta}{2}n}\) on a quantum computer.


Success Probability Block Cipher Satisfying Assignment Depth Circuit Exact Complexity 
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  1. [BBHT96]
    Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching (May 1996)Google Scholar
  2. [Cal08]
    Calabro, C.: A lower bound on the size of series-parallel graphs dense in long paths. In: Electronic Colloquium on Computational Complexity (ECCC), vol. 15(110) (2008)Google Scholar
  3. [CIKP08]
    Calabro, C., Impagliazzo, R., Kabanets, V., Paturi, R.: The complexity of unique k-sat: An isolation lemma for k-cnfs. J. Comput. Syst. Sci. 74(3), 386–393 (2008)CrossRefzbMATHMathSciNetGoogle Scholar
  4. [CIP06]
    Calabro, C., Impagliazzo, R., Paturi, R.: A duality between clause width and clause density for sat. In: CCC 2006: Proceedings of the 21st Annual IEEE Conference on Computational Complexity, Washington, DC, USA, 2006, pp. 252–260. IEEE Computer Society Press, Los Alamitos (2006)Google Scholar
  5. [DW05]
    Dantsin, E., Wolpert, A.: An improved upper bound for sat. In: Bacchus, F., Walsh, T. (eds.) SAT 2005. LNCS, vol. 3569, pp. 400–407. Springer, Heidelberg (2005)Google Scholar
  6. [Gro96]
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: STOC, pp. 212–219 (1996)Google Scholar
  7. [IP01]
    Impagliazzo, R., Paturi, R.: On the complexity of k-sat. J. Comput. Syst. Sci. 62(2), 367–375 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  8. [IPZ01]
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? J. Comput. Syst. Sci. 63(4), 512–530 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  9. [Nur09]
    Nurik, S.: Personal communication. To appear in ECCC (2009)Google Scholar
  10. [PPSZ05]
    Paturi, R., Pudlák, P., Saks, M.E., Zane, F.: An improved exponential-time algorithm for k-sat. J. ACM 52(3), 337–364 (2005)CrossRefMathSciNetGoogle Scholar
  11. [PPZ99]
    Paturi, R., Pudlák, P., Zane, F.: Satisfiability coding lemma. Chicago Journal of Theoretical Computer Science 115 ( December 1999)Google Scholar
  12. [PSZ00]
    Paturi, R., Saks, M.E., Zane, F.: Exponential lower bounds for depth 3 boolean circuits. Computational Complexity 9(1), 1–15 (2000); Preliminary version in 29th annual ACM Symposium on Theory of Computing, pp. 96–91 (1997)CrossRefzbMATHMathSciNetGoogle Scholar
  13. [Sch99]
    Schöning, U.: A probabilistic algorithm for k-sat and constraint satisfaction problems. In: FOCS, pp. 410–414 (1999)Google Scholar
  14. [Sch05]
    Schuler, R.: An algorithm for the satisfiability problem of formulas in conjunctive normal form. J. Algorithms 54(1), 40–44 (2005)CrossRefzbMATHMathSciNetGoogle Scholar
  15. [Val77]
    Valiant, L.: Graph-theoretic arguments in low level complexity. In: Gruska, J. (ed.) MFCS 1977. LNCS, vol. 53, Springer, Heidelberg (1977)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Chris Calabro
    • 1
  • Russell Impagliazzo
    • 1
  • Ramamohan Paturi
    • 1
  1. 1.Department of Computer Science and EngineeringUniversity of CaliforniaSan Diego, La JollaUSA

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