A superpolynomial lower bound for a circuit computing the clique function with at most (1/6) log log n negation gates

Abstract

We investigate about a lower bound on the size of 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 the three approaches: the restriction applied for constant depth circuits[Has], the approximation method applied for monotone circuits[Raz2] and boundary covering developed in the present paper. Based on the approach the following statement is established: If a circuit C with at most ⌈(1/6) log log m⌋ negation gates detects cliques of size $(\log m)^{3(\log {\mathbf{ }}m)^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} }$ in a graph with m vertices, then C contains at least $2^{({\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 5$}})(\log {\mathbf{ }}m)^{(\log {\mathbf{ }}m)^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} } }$ gates. In addition, we present a general relationship between negation-limited circuit size and monotone circuit size of an arbitrary monotone function.