, Volume 7, Issue 1, pp 1-22

The monotone circuit complexity of boolean functions

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Recently, Razborov obtained superpolynomial lower bounds for monotone circuits that cliques in graphs. In particular, Razborov showed that detecting cliques of sizes in a graphm vertices requires monotone circuits of size Ω(m s /(logm)2s ) for fixeds, and sizem Ω(logm) form/4].

In this paper we modify the arguments of Razborov to obtain exponential lower bounds for circuits. In particular, detecting cliques of size (1/4) (m/logm)2/3 requires monotone circuits exp (Ω((m/logm)1/3)). For fixeds, any monotone circuit that detects cliques of sizes requiresm) s ) AND gates. We show that even a very rough approximation of the maximum clique of a graph requires superpolynomial size monotone circuits, and give lower bounds for some Boolean functions. Our best lower bound for an NP function ofn variables is exp (Ω(n 1/4 · (logn)1/2)), improving a recent result of exp (Ω(n 1/8-ε)) due to Andreev.

First author supported in part by Allon Fellowship, by Bat Sheva de-Rotschild Foundation by the Fund for basic research administered by the Israel Academy of Sciences.
Second author supported in part by a National Science Foundation Graduate Fellowship.