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

• Kazuyuki Amano
• Akira Maruoka
Contributed Papers Circuit Complexity
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1450)

## 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.

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