A Superpolynomial Lower Bound for a Circuit Computing the Clique Function with at most (1/6)log log n Negation Gates

Abstract

In this paper, we investigate the lower bound on the number of gates in a Boolean circuit that computes the clique function with a limited number of negation gates. To derive strong lower bounds on the size of such a circuit we develop a new approach by combining three approaches: the restriction applied to constant depth circuits due to Hastad, the approximation method applied to monotone circuits due to Razborov, and the boundary covering developed in the present paper. We prove that if a circuit $C$ with at most $\floor{(1/6) \log \log m}$ negation gates detects cliques of size $(\log m)^{3(\log m)^{1/2}}$ in a graph with $m$ vertices, then $C$ contains at least $2^{(1/5)(\log m)^{(\log m)^{1/2}}}$ gates. No nontrivial lower bounds on the size of such circuits were previously known, even if we restrict the number of negation gates to be a constant. Moreover, it follows from a result of Fischer [{\it Lect. Notes Comput. Sci.,} 33 (1974), pp. 71--82] that if one can improve the number of negation gates from $\floor{(1/6)\log\log m}$ to $\floor{2\log m}$ in the statement, then we have P $\neq$ NP. We also show that the problem of lower bounding the negation-limited circuit complexity can be reduced to the one of lower bounding the maximum of the monotone circuit complexity of the functions in a certain class of monotone functions.