Minimum propositional proof length is NP-hard to linearly approximate

Extended abstract
  • Michael Alekhnovich
  • Sam Buss
  • Shlomo Moran
  • Toniann Pitassi
Contributed Papers Complexity of Hard Problems
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1450)


We prove that the problem of determining the minimum propositional proof length is NP-hard to approximate within any constant factor. These results hold for all Frege systems, for all extended Frege systems, for resolution and Horn resolution, and for the sequent calculus and the cut-free sequent calculus. Also, if NP is not in \(QP = DTIME(n^{log^{O(1)} n} )\), then it is impossible to approximate minimum propositional proof length within a factor of \(2^{log^{(1 - \varepsilon )} n}\) for any є > 0. All these hardness of approximation results apply to proof length measured either by number of symbols or by number of inferences, for tree-like or dag-like proofs. We introduce the Monotone Minimum (Circuit) Satisfying Assignment problem and prove the same hardness results for Monotone Minimum (Circuit) Satisfying Assignment.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Arora, L. Babai, J. Stern, and Z. Sweedyk, The hardness of approximate optima in lattices, codes, and systems of linear equations, Journal of Computer and System Sciences, 54 (1997), pp. 317–331. Earlier version in Proc. 34th Symp. Found. of Comp. Sci., 1993, pp.724–733.MATHMathSciNetCrossRefGoogle Scholar
  2. 2.
    S. Arora and C. Lund, Hardness of approximations, in Approximation Algorithms for NP-hard Problems, D. S. Hochbaum, ed., PWS Publishing Co., Boston, 1996, p. ???Google Scholar
  3. 3.
    P. Beame and T. Pitassi, Simplified and improved resolution lower bounds, in Proceedings, 37th Annual Symposium on Foundations of Computer Science, Los Alamitos, California, 1996, IEEE Computer Society, pp. 274–282.Google Scholar
  4. 4.
    M. Bellare, Proof checking and approximation: Towards tight results, SIGACT News, 27 (1996), pp. 2–13. Revised version at Scholar
  5. 5.
    M. Bellare, S. Goldwasser, C. Lund, and A. Russell, Efficient probabalistically checkable proofs and applications to approximation, in Proceedings of the Twenty-Fifth Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, 1993, pp. 294–304.Google Scholar
  6. 6.
    M. L. Bonet, T. Pitassi, and R. Raz, No feasible interpolation for TC0-Frege proofs, in Proceedings of the 38th Annual Symposium on Foundations of Computer Science, Piscataway, New Jersey, 1997, IEEE Computer Society, pp. 264–263.Google Scholar
  7. 7.
    S. R. Buss, On Gödel's theorems on lengths of proofs II: Lower bounds for recognizing k symbol provability, in Feasible Mathematics II, P. Clote and J. Remmel, eds., Birkhäauser-Boston, 1995, pp. 57–90.Google Scholar
  8. 8.
    M. Clegg, J. Edmonds, and R. Impagliazzo, Using the Groebner basis algorithm to find proofs of unsatisfiability, in Proceedings of the Twenty-eighth Annual ACM Symposium on the Theory of Computing, Association for Computing Machinery, 1996, pp. 174–183.Google Scholar
  9. 9.
    U. Feige, A threshold of ln n for approximating set cover, in Proceedings of the Twenty-Eighth Annual ACM Symposium on Theory of Computing, Association for Computing Machinery, 1996, pp. 314–318.Google Scholar
  10. 10.
    K. Iwama, Complexity of finding short resolution proofs, in Mathematical Foundations of Computer Science 1997, I. Prívara and P. Ruzicka, eds., Lecture Notes in Computer Science #1295, Springer-Verlag, 1997, pp. 309–318.Google Scholar
  11. 11.
    K. Iwama and E. Miyano, Intractibility of read-once resolution, in Proceedings of the Tenth Annual Conference on Structure in Complexity Theory, Los Alamitos, California, 1995, IEEE Computer Society, pp. 29–36.Google Scholar
  12. 12.
    S. Khanna, M. Sudan, and L. Trevisan, Constraint satisfaction: The approximability of minimization problems, in Twelfth Annual Conference on Computational Complexity, IEEE Computer Society, 1997, pp. 282–296.Google Scholar
  13. 13.
    J. Krajíček and P. Pudlák, Some consequences of cryptographic conjectures for S21 and EF, in Logic and Computational Complexity, D. Leivant, ed., Berlin, 1995, Springer-Verlag, pp. 210–220.Google Scholar
  14. 14.
    C. Lund and M. Yannakakis, On the hardness of approximating minimization problems, Journal of the Association for Computing Machinery, 41 (1994), pp. 960–981.MATHMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Michael Alekhnovich
    • 1
  • Sam Buss
    • 2
  • Shlomo Moran
    • 3
  • Toniann Pitassi
    • 4
  1. 1.Moscow State UniversityRussia
  2. 2.University of CaliforniaSan Diego
  3. 3.TechnionIsrael Institute of TechnologyIsrael
  4. 4.University of ArizonaTucson

Personalised recommendations