The monotone circuit complexity of boolean functions |
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Authors: | Noga Alon Ravi B Boppana |
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Institution: | (1) Department of Mathematics, Tel Aviv University, Tel Aviv, Israel;(2) IBM Almaden Research Center, 650 Harry Road, 95120 San Jose, CA, USA;(3) Laboratory for Computer Science, Massachusetts Inst. of Tech., 545 Technology Square, 02139 Cambridge, Mass., USA |
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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.
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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Keywords: | 68 E 10 68 C 25 |
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