An Optimal Lower Bound for Resolution with 2-Conjunctions

  • Jan Johannsen
  • N. S. Narayanaswamy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2420)


A lower bound is proved for refutations of certain clause sets in a generalization of Resolution that allows cuts on conjunctions of width 2. The hard clauses are the Tseitin graph formulas for a class of logarithmic degree expander graphs. The bound is optimal in the sense that it is truly exponential in the number of variables.


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  1. 1.
    A. Atserias and M. L. Bonet. On the automatizability of resolution and related propositional proof systems. Technical Report ECCC TR02-010, Electronic Colloquium on Computational Complexity, 2002.Google Scholar
  2. 2.
    A. Atserias, M. L. Bonet, and J. L. Esteban. Lower bounds for the weak pigeonhole principle beyond resolution, 2002. To appear in Information and Computation. Preliminary Version in Proc. 28th International Colloquium on Automata Languages and Programming, 2001.Google Scholar
  3. 3.
    R. Beigel and D. Eppstein. 3-coloring in time O(1.3446n): a no-MIS algorithm. In Proc. 36th IEEE Symposiom on Foundations of Computer Science, pages 444–452, 1995.Google Scholar
  4. 4.
    E. Ben-Sasson. Hard examples for bounded-depth Frege. To appear in Proc. 34th ACM Symposium on Theory of Computing, 2002.Google Scholar
  5. 5.
    E. Ben-Sasson and A. Wigderson. Short proofs are narrow — resolution made simple. Journal of the ACM, 48:149–169, 2001. Preliminary Version in Proc. 31st Symposium on Theory of Computing, 1999.MATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    B. Bollobás. Random Graphs. Academic Press, 1985.Google Scholar
  7. 7.
    J. Krajíček. On the weak pigeonhole principle. Fundamenta Mathematicae, 170:123–140, 2001.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    N. Segerlind, S. R. Buss, and R. Impagliazzo. A switching lemma for small restrictions and lower bounds for k-DNF resolution. Submitted for publication, 2002.Google Scholar
  9. 9.
    G. Tseitin. On the complexity of derivation in propositional calculus. In A. O. Slisenko, editor, Studies in Constructive Mathematics and Mathematical Logic, Part 2, pages 115–125. Consultants Bureau, 1970.Google Scholar
  10. 10.
    A. Urquhart. Hard examples for resolution. Journal of the ACM, 34:209–219, 1987.MATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Jan Johannsen
    • 1
  • N. S. Narayanaswamy
    • 1
  1. 1.Institut für InformatikLudwig-Maximilians-Universität MünchenGermany

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