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The monotone circuit complexity of Boolean functions (1987)
| Venue: | COMBINATORICA |
| Citations: | 144 - 2 self |
BibTeX
@ARTICLE{Alon87themonotone,
author = {N. Alon and R. B. Boppana},
title = {The monotone circuit complexity of Boolean functions},
journal = {COMBINATORICA},
year = {1987},
volume = {7},
number = {1},
pages = {1--22}
}
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Abstract
Recently, Razborov obtained superpolynomial lower bounds for monotone circuits that lect cliques in graphs. In particular, Razborov showed that detecting cliques of size s in a graph dh m vertices requires monotone circuits of size.Q(m-'/(log m) ~') for fixed s, and size rn ao°~') for,.:[log ml4J. In this paper we modify the arguments of Razborov to obtain exponential lower bounds for monotone circuits. In particular, detecting cliques of size (1/4) (m/log m) ~'/a requires monotone circuits f size exp (£2((m/log m)~/:~)). For fixed s, any monotone circuit that detects cliques of size s requires 'm'/(log m)') AND gates. We show that even a very rough approximation of the maximum clique e of a graph requires superpolynomial size monotone circuits, and give lower bounds for some net Boolean functions. Our best lower bound fi~r an NP function of n variables is exp (f2(n w4. (log n)~/~)), improving a recent result of exp (f2(nws-')) due to Andreev.
