A tighter relation between sensitivity complexity and certificate complexity

Abstract

The sensitivity conjecture, proposed by Nisan and Szegedy in which asserts that for any Boolean function, its sensitivity complexity is polynomially related to the block sensitivity complexity, is one of the most important and challenging problems in the study of decision tree complexity. Despite of a lot of efforts, the best known upper bounds of block sensitivity, as well as the certificate complexity, is still exponential in terms of sensitivity paper, we give a better upper bound for certificate complexity and block sensitivity, (formula presented) where bs(f),C(f) and s(f) are the block sensitivity, certificate complexity and sensitivity, respectively. The proof is based on a deep investigation on the structure of the sensitivity graph. We also provide a tighter relationship between the 0-certificate complexity C:0(f) and 0-sensitivity (formula presented) for functions with small 1-sensitivity(formula presented).