Advertisement

computational complexity

, Volume 20, Issue 4, pp 597–614 | Cite as

Lower Bounds for k-DNF Resolution on Random 3-CNFs

  • Michael Alekhnovich
Article

Abstract

We prove exponential lower bounds on refutations of a random 3-CNF with linear number of clauses by k-DNF Resolution for \({k\leq \sqrt{\log n/\log\log n}}\) . For this we design a specially tailored random restrictions that preserve the structure of the input random 3-CNF while mapping every k-DNF with large covering number to one with high probability. Next we make use of the switching lemma for small restrictions by Segerlind, Buss and Impagliazzo to prove the lower bound.

This work improves the previously known lower bound for Res(2) system on random 3-CNFs by Atserias, Bonet and Esteban and the result of Segerlind, Buss, Impagliazzo stating that random O(k 2)-CNF do not possess short Res(k) refutations.

Keywords

Propositional proof complexity random CNFs 

Subject classification

68Q17 68T15 03B70 03F20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Alekhnovich M., Ben-Sasson E., Razborov A., Wigderson A. (2004) Pseudorandom generators in propositional proof complexity. SIAM Journal on Computing 34(1): 67–88zbMATHCrossRefMathSciNetGoogle Scholar
  2. M. Alekhnovich, A. Borodin, J. Buresh-Oppenheim, R. Impgaliazzo, A. Magen & T. Pitassi (2011). Toward a model for backtracking and dynamic programming. Computational Complexity in this volume.Google Scholar
  3. Alekhnovich M., Hirsch E., Itsykson D. (2005) Exponential lower bounds for the running time of DPLL algorithms on satisfiable formulas. Journal of Automated Reasoning 35(1–3): 51–72zbMATHMathSciNetGoogle Scholar
  4. Alekhnovich M., Razborov A. (2003) Lower bounds for the polynomial calculus: non-binomial case. Proceedings of the Steklov Institute of Mathematics 242: 18–35MathSciNetGoogle Scholar
  5. Atserias A., Bonet M.L., Esteban J.L. (2002) Lower bounds for the weak pigeonhole principle beyond resolution. Information and Computation 176: 136–152zbMATHCrossRefMathSciNetGoogle Scholar
  6. Beame P., Karp R., Pitassi T., Saks M. (2002) The efficiency of resolution and Davis-Putnam procedures. SIAM Journal on Computing 31(4): 1048–1075zbMATHCrossRefMathSciNetGoogle Scholar
  7. E. Ben-Sasson & R. Impagliazzo (1999). Random CNF’ s are Hard for the Polynomial Calculus. In Proceedings of the 40th IEEE FOCS, 415–421.Google Scholar
  8. Ben-Sasson E., Wigderson A. (2001) Short Proofs are Narrow—Resolution made Simple. Journal of the ACM 48(2): 149–169zbMATHCrossRefMathSciNetGoogle Scholar
  9. Chvátal V., Szemerédi E. (1988) Many hard examples for resolution. Journal of the ACM 35(4): 759–768zbMATHCrossRefGoogle Scholar
  10. U. Feige (2002). Relations between Average Case Complexity and Approximation Complexity. In Proceedings of the 34th ACM Symposium on the Theory of Computing, 534–543.Google Scholar
  11. Friedgut E. (1999) Sharp thresholds of graph properties. Journal of the American Math. Society 12(4): 1017–1054zbMATHCrossRefMathSciNetGoogle Scholar
  12. Krajíček J. (2001) On the weak pigeonhole principle. Fundamenta Mathematicae 170((1–3): 123–140zbMATHCrossRefMathSciNetGoogle Scholar
  13. Segerlind N., Buss S., Impagliazzo R. (2004) A Switching Lemma for Small Restrictions and Lower Bounds for k-DNF Resolution. SIAM Journal on Computing 33(5): 1171–1200zbMATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • Michael Alekhnovich
    • 1
  1. 1.Institute for Advanced StudyPrincetonUSA

Personalised recommendations