Summary: COMBINATOR1CA7 (1) (1987) 1--22
THE MONOTONE CIRCUIT COMPLEXITY
OF BOOLEAN FUNCTIONS
N. ALON and R. B. BOPPANA
Received 15 November 1985
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-'/(logm)~') for fixed s, and size rnao°~') for
,. :[log ml4J.
In this paper we modify the arguments of Razborov to obtain exponential lower bounds for
,motone circuits. In particular, detectingcliques of size (1/4) (m/log m)~'/arequires monotone circuits
f size exp (£2((m/log m)~/:~)).For fixed s, any inonotone 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(nw4.
(log n)~/~)),improving a recent result of exp (f2(nws-')) due to Andreev.
In 1949, Shannon  showed that almost all Boolean functions have expo-
·ntially large circuit complexity. Unfortunately, the best circuit lower bound for a
oblem in NP is only 311 (Blum ). Circuit lower bounds are important since a