Combinatorica

, Volume 8, Issue 1, pp 141–142 | Cite as

The gap between monotone and non-monotone circuit complexity is exponential

  • É. Tardos
Notes

Abstract

A. A. Razborov has shown that there exists a polynomial time computable monotone Boolean function whose monotone circuit complexity is at leastnc losn. We observe that this lower bound can be improved to exp(cn1/6−o(1)). The proof is immediate by combining the Alon—Boppana version of another argument of Razborov with results of Grötschel—Lovász—Schrijver on the Lovász — capacity, ϑ of a graph.

AMS subject classification

68 C 25 

References

  1. [1]
    N. Alon andR. Boppana, The monotone circuit complexity of Boolean functions,Combinatorica 7 (1987), 1–23.Google Scholar
  2. [2]
    M. Grötschler, L. Lovász andA. Schrijver, The ellipsoid method and its consequences in combinatorial optimization,Combinatorica 1 (1981), 169–197.Google Scholar
  3. [3]
    G. L. Khachiyan, A polynomial algorithm in linear programming,Doklady Akademii Nauk SSSR 244 (1979), 1093–1096 (English translation:Soviet Math. Dokl. 20, 191–194).Google Scholar
  4. [4]
    L. Lovász, On the Shannon capacity of a graph,IEEE Trans. on Information Theory 25 (1979), 1–7.Google Scholar
  5. [5]
    L. Lovász, An Algorithmic Theory of Numbers, Graphs and Convexity, SIAM Philadelphia 1986.Google Scholar
  6. [6]
    A. A. Razborov, Lower bounds on the monotone complexity of some Boolean functions,Doklady Akademii Nauk SSSR 281 (1985), 798–801.Google Scholar
  7. [7]
    A. A. Razborov, A lower bound on the monotone network complexity of the logical permanent,Matematischi Zametki 37 (1985), 887–900.Google Scholar

Copyright information

© Akadémiai Kiadó 1988

Authors and Affiliations

  • É. Tardos
    • 1
  1. 1.Comp. Sci. Dept.Eötvös UniversityBudapestHungary

Personalised recommendations