Proceedings of 37th Conference on Foundations of Computer Science (1996)
Oct. 14, 1996 to Oct. 16, 1996
K. Amano , Graduate Sch. of Inf. Sci., Tohoku Univ., Sendai, Japan
A. Maruoka , Graduate Sch. of Inf. Sci., Tohoku Univ., Sendai, Japan
Developing some techniques for the approximation method, we establish precise versions of the following statements concerning lower bounds for circuits that detect cliques of size s in a graph with m vertices. For 5/spl les/s/spl les/m/4, a monotone circuit computing CLIQUE(m, s) contains at least (1/2) 1.8/sup min(/spl radic/s-1/2,m/(4s))/ gates. If a non-monotone circuit computes CLIQUE using a "small" amount of negation, then the circuit contains an exponential number of gates. The former is proved very simply using so called bottleneck counting argument within the framework of approximation, whereas the latter is verified introducing a notion of restricting negation and generalizing the sunflower contraction.
Boolean functions; approximation method; precise versions; lower bounds; cliques; vertices; monotone circuit; bottleneck counting argument; sunflower contraction
K. Amano and A. Maruoka, "Potential of the approximation method," Proceedings of 37th Conference on Foundations of Computer Science(FOCS), Burlington, VT, 1996, pp. 431.