Improved Lower Bounds for Tree-Like Resolution over Linear Inequalities

  • Arist Kojevnikov
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4501)


We continue a study initiated by Krajíček of a Resolution-like proof system working with clauses of linear inequalities, R(CP). For all proof systems of this kind Krajíček proved in [1] an exponential lower bound of the form:
$$ \frac{exp(n^{\Omega(1)})}{M^{O(W\log^2 n)}}\enspace, $$
where M is the maximal absolute value of coefficients in a given proof and W is the maximal clause width.

In this paper we improve this lower bound. For tree-like R(CP)-like proof systems we remove a dependence on the maximal absolute value of coefficients M, hence, we give the answer to an open question from [2]. Proof follows from an upper bound on the real communication complexity of a polyhedra.


propositional proof complexity integer programming cutting planes 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Krajíček, J.: Discretely ordered modules as a first-order extension of the cutting planes proof system. Journal of Symbolic Logic 63(4), 1582–1596 (1998)CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    Krajíček, J.: Interpolation by a game. Mathematical Logic Quarterly 44(40), 450–458 (1998)zbMATHGoogle Scholar
  3. 3.
    Land, H., Doig, A.G.: An automatic method for solving discrete programming problems. Econometrica 28, 497–520 (1960)CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    Bonet, M., Pitassi, T., Raz, R.: Lower bounds for cutting planes proofs with small coefficients. The Journal of Symbolic Logic 62(3), 708–728 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Krajíček, J.: On the weak pigeonhole principle. Fundamenta Mathematicæ 170(1-3), 123–140 (2001)zbMATHCrossRefGoogle Scholar
  6. 6.
    Atserias, A., Bonet, M.L., Esteban, J.L.: Lower bounds for the weak pigeonhole principle and random formulas beyond resolution. Information and Computation 176(2), 136–152 (2002)CrossRefMathSciNetzbMATHGoogle Scholar
  7. 7.
    Segerlind, N., Buss, S.R., Impagliazzo, R.: A Switching Lemma for Small Restrictions and Lower Bounds for k-DNF Resolution. SIAM Journal on Computing 33(5), 1171–1200 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  8. 8.
    Alekhnovich, M.: Lower bounds for k-DNF resolution on random 3-CNFs. In: STOC ’05: Proceedings of the thirty-seventh annual ACM symposium on Theory of computing, pp. 251–256. ACM Press, New York (2005)CrossRefGoogle Scholar
  9. 9.
    Prestwich, S.: Incomplete dynamic backtracking for linear pseudo-boolean problems. Annals of Operations Research 130, 57–73 (2004)CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    Chai, D., Kuehlmann, A.: A fast pseudo-boolean constraint solver. IEEE Trans. on CAD of Integrated Circuits and Systems 24(3), 305–317 (2005)CrossRefGoogle Scholar
  11. 11.
    Manquinho, V.M., Marques-Silva, J.: On using cutting planes in pseudo-boolean optimization. Journal of Satisfiability, Boolean Modeling and Computation 2, 209–219 (2006)zbMATHGoogle Scholar
  12. 12.
    Razborov, A.A.: Lower bounds on the monotone complexity of some Boolean functions (in Russian). Dokl. Akad. Nauk SSSR 281(4), 354–357 (1985), English translation in: Soviet Math. Dokl. 31, 354–357 (1985)MathSciNetGoogle Scholar
  13. 13.
    Karchmer, M., Wigderson, A.: Monotone circuits for connectivity require super-logarithmic depth. SIAM Journal on Discrete Mathematics 3(2), 255–265 (1990)CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Krajíček, J.: Interpolation theorems, lower bounds for proof systems, and independence results for bounded arithmetic. Journal of Symbolic Logic 62(2), 457–486 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Kushilevitz, E., Nisan, N.: Communication Complexity. Cambridge University Press, Cambridge (1997)zbMATHGoogle Scholar
  16. 16.
    Bonet, M.L., et al.: On the relative complexity of resolution refinements and cutting planes proof systems. SIAM J. Comp. 30(5), 1462–1484 (2000)CrossRefMathSciNetzbMATHGoogle Scholar
  17. 17.
    Pudlák, P.: Lower bounds for resolution and cutting plane proofs and monotone computations. Journal of Symbolic Logic 62(3), 981–998 (1997)CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Atserias, A., Bonet, M.L.: On the automatizability of resolution and related propositional proof systems. Information and Computation 189(2), 182–201 (2004)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer Berlin Heidelberg 2007

Authors and Affiliations

  • Arist Kojevnikov
    • 1
  1. 1.St.Petersburg Department of Steklov Institute of Mathematics, 27 Fontanka, 191023 St.PetersburgRussia

Personalised recommendations