Abstract
The sensitivity conjecture, proposed by Nisan and Szegedy in 1994 [20] 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 [1, 5]. In this paper, we give a better upper bound for certificate complexity and block sensitivity, \(bs(f)\le C(f)\le (\frac{8}{9} + o(1))s(f)2^{s(f) - 1}\), 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 \(s_0(f)\) for functions with small 1-sensitivity \(s_1(f)\).
This work was supported in part by the National Natural Science Foundation of China Grant 61433014, 61502449, 61602440, the 973 Program of China Grants No. 2016YFB1000201 and the China National Program for support of Top-notch Young Professionals.
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Notes
- 1.
The function p can be viewed as a vector, and we sometimes use \(p_i\) to represent p(i).
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He, K., Li, Q., Sun, X. (2017). A Tighter Relation Between Sensitivity Complexity and Certificate Complexity. In: Cao, Y., Chen, J. (eds) Computing and Combinatorics. COCOON 2017. Lecture Notes in Computer Science(), vol 10392. Springer, Cham. https://doi.org/10.1007/978-3-319-62389-4_22
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