Abstract
It is known that, for every constant k ⩾ 3 , the presence of a k-clique (a complete sub-graph on k vertices) in an n-vertex graph cannot be detected by a monotone boolean circuit using much fewer than n k gates. We show that, for every constant k, the presence of an ( n - k ) -clique in an n-vertex graph can be detected by a monotone circuit using only a logarithmic number of fanin-2 OR gates; the total number of gates does not exceed O ( n 2 log n ) . Moreover, if we allow unbounded fanin, then a logarithmic number of gates is enough.
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