Potential of the Approximation Method

Since Razborov, based on the approximation method, succeeded to obtain a superpolynomial lower bound on the size of monotone circuits computing the clique function, much effort has been devoted to explore the method and derive good lower bounds[K, NM, R1, R2, RR]. Employing the approximation method, Alon and Boppana[AB] obtained an exponential lower bound for monotone circuits computing the clique function. Using seemingly different argument, called bottleneck counting, Haken[H] derived an exponential lower bound for monotone circuits computing a variant of the clique function. Furthermore, it was shown that the extended approximation method, in principle, can provide tight lower bounds for non-monotone circuits[R2]. In spite of these progress in proving lower bounds, the question of how to apply the method to obtain good lower bounds has remained largely unknown. In this paper, we explore further the method in two ways. First, we use the bottleneck counting argument within the framework of approximation so as to obtain exponential lower bounds for monotone circuits computing the clique function as well as some other function defined in terms of polynomials. Denoting the clique function detecting cliques of size s in a graph with m vertices by CLIQUE(m;s), our lower bound for monotone circuits computing CLIQUE(m; s) is given by (1=2)1:8min( p s 1=2;m=(4s)) for 5 s m=4. The best lower bound obtained so far for the clique function is (1=8)(m=(4s3=2 logm)) p