The monotone circuit complexity of boolean functions

Abstract

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.

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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.

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Alon, N., Boppana, R.B. The monotone circuit complexity of boolean functions. Combinatorica 7, 1–22 (1987). https://doi.org/10.1007/BF02579196

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AMS subject classification (1980)

  • 68 E 10
  • 68 C 25