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

Abstract

A. A. Razborov has shown that there exists a polynomial time computable monotone Boolean function whose monotone circuit complexity is at leastn c losn. We observe that this lower bound can be improved to exp(cn 1/6−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.

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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 

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Tardos, É. The gap between monotone and non-monotone circuit complexity is exponential. Combinatorica 8, 141–142 (1988). https://doi.org/10.1007/BF02122563

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  • 68 C 25