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


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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Tardos, É. The gap between monotone and non-monotone circuit complexity is exponential. Combinatorica 8, 141–142 (1988).

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