敏感性与证书复杂性之间的更紧密关系
A Tighter Relation between Sensitivity and Certificate Complexity
摘要：
灵敏度猜想是决策树复杂性理论中最重要也是最具挑战性的问题之一，它认为灵敏度复杂性与块灵敏度复杂性呈多项式关系。尽管做了很多努力，块敏感性的最佳已知上限以及证书复杂性在敏感性方面仍然是指数级的：$bs(f)\leq C(f)\leq\max\{2^{s(f)-1}(s(f)-\frac{1}{3}),s(f)\}$。本文给出了$bs(f)\leq C(f)\leq(\frac{8}{9} + o(1))s(f)2^{s(f) - 1}$的一个较好的上界。证明是基于对灵敏度图结构的深入研究。我们还为带有$s_1(f)=2$的函数提供了$C_0(f)$和$s_0(f)$之间更紧密的关系。
英文摘要：
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 $s_1(f)=2$.
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