A Tighter Relation between Sensitivity and Certificate Complexity

The sensitivity conjecture which claims that the sensitivity complexity is polynomially related to block sensitivity complexity, is one of the most important and challenging problem in decision tree complexity theory. Despite of a lot of efforts, the best known upper bound of block sensitivity, as well as the certificate complexity, are still exponential in terms of sensitivity: $bs(f)\leq C(f)\leq\max\{2^{s(f)-1}(s(f)-\frac{1}{3}),s(f)\}$. In this paper, we give a better upper bound of $bs(f)\leq C(f)\leq(\frac{8}{9} + o(1))s(f)2^{s(f) - 1}$. The proof is based on a deep investigation on the structure of the sensitivity graph. We also provide a tighter relationship between $C_0(f)$ and $s_0(f)$ for functions with $s_1(f)=2$.