On the Limits of Sparsification

  • Rahul Santhanam
  • Srikanth Srinivasan
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)


Impagliazzo, Paturi and Zane (JCSS 2001) proved a sparsification lemma for k-CNFs: every k-CNF is a sub-exponential size disjunction of k-CNFs with a linear number of clauses. This lemma has subsequently played a key role in the study of the exact complexity of the satisfiability problem. A natural question is whether an analogous structural result holds for CNFs or even for broader non-uniform classes such as constant-depth circuits or Boolean formulae. We prove a very strong negative result in this connection: For every superlinear function f(n), there are CNFs of size f(n) which cannot be written as a disjunction of 2n − εn CNFs each having a linear number of clauses for any ε > 0. We also give a hierarchy of such non-sparsifiable CNFs: For every k, there is a k′ for which there are CNFs of size nk which cannot be written as a sub-exponential size disjunction of CNFs of size nk. Furthermore, our lower bounds hold not just against CNFs but against an arbitrary family of functions as long as the cardinality of the family is appropriately bounded.

As by-products of our result, we make progress both on questions about circuit lower bounds for depth-3 circuits and satisfiability algorithms for constant-depth circuits. Improving on a result of Impagliazzo, Paturi and Zane, for any f(n) = ω(n log(n)), we define a pseudo-random function generator with seed length f(n) such that with high probability, a function in the output of this generator does not have depth-3 circuits of size 2n − o(n) with bounded bottom fan-in. We show that if we could decrease the seed length of our generator below n, we would get an explicit function which does not have linear-size logarithmic-depth series-parallel circuits, solving a long-standing open question.

Motivated by the question of whether CNFs sparsify into bounded-depth circuits, we show a simplification result for bounded-depth circuits: any bounded-depth circuit of linear size can be written as a sub-exponential size disjunction of linear-size constant-width CNFs. As a corollary, we show that if there is an algorithm for CNF satisfiability which runs in time O(2αn) for some fixed α < 1 on CNFs of linear size, then there is an algorithm for satisfiability of linear-size constant-depth circuits which runs in time O(2(α + o(1))n).


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arora, S., Barak, B.: Computational Complexity - A Modern Approach. Cambridge University Press (2009)Google Scholar
  2. 2.
    Calabro, C., Impagliazzo, R., Paturi, R.: A duality between clause width and clause density for SAT. In: Proceedings of IEEE Conference on Computational Complexity, pp. 252–260 (2006)Google Scholar
  3. 3.
    Impagliazzo, R., Matthews, W., Paturi, R.: A satisfiability algorithm for AC0. In: Proceedings of Symposium on Discrete Algorithms (to appear, 2012)Google Scholar
  4. 4.
    Impagliazzo, R., Paturi, R.: On the complexity of k-sat. Journal of Computer and System Sciences 63(4), 512–530 (2001)MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    Impagliazzo, R., Paturi, R., Zane, F.: Which problems have strongly exponential complexity? Journal of Computer and System Sciences 62(4), 512–530 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Malik, S., Zhang, L.: Boolean satisfiability from theoretical hardness to practical success. Communications of the ACM 52(8), 76–82 (2009)CrossRefGoogle Scholar
  7. 7.
    Miltersen, P.B., Radhakrishnan, J., Wegener, I.: On converting cnf to dnf. Theoretical Computer Science 347(1-2), 325–335 (2005)MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Paturi, R., Pudlak, P., Saks, M., Zane, F.: An improved exponential-time algorithm for k-sat. In: Proceedings of 39th International Symposium on Foundations of Computer Sciece (FOCS), pp. 628–637 (1998)Google Scholar
  9. 9.
    Paturi, R., Pudlak, P., Zane, F.: Satisfiability coding lemma. In: Proceedings of 38th International Symposium on Foundations of Computer Science (FOCS), pp. 566–574 (1997)Google Scholar
  10. 10.
    Santhanam, R.: Fighting perebor: New and improved algorithms for formula and QBF satisfiability. In: Proceedings of 51st Annual IEEE Symposium on Foundations of Computer Science, pp. 183–192 (2010)Google Scholar
  11. 11.
    Santhanam, R., Srinivasan, S.: On the limits of sparsification. Electronic Colloquium on Computational Complexity (ECCC) 18, 131 (2011)Google Scholar
  12. 12.
    Schuler, R.: An algorithm for the satisfiability problem of formulas in conjunctive normal form. J. Algorithms 54(1), 40–44 (2005)MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Valiant, L.G.: Graph-Theoretic Arguments in Low-Level Complexity. In: Gruska, J. (ed.) MFCS 1977. LNCS, vol. 53, pp. 162–176. Springer, Heidelberg (1977)CrossRefGoogle Scholar
  14. 14.
    Williams, R.: Improving exhaustive search implies superpolynomial lower bounds. In: Proceedings of the 42nd Annual ACM Symposium on Theory of Computing, pp. 231–240 (2010)Google Scholar
  15. 15.
    Williams, R.: Non-uniform ACC circuit lower bounds. In: Proceedings of 26th Annual IEEE Conference on Computational Complexity, pp. 115–125 (2011)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Rahul Santhanam
    • 1
  • Srikanth Srinivasan
    • 2
  1. 1.University of EdinburghUK
  2. 2.DIMACSRutgers UniversityUSA

Personalised recommendations