块敏感性和证书复杂性的新上限
New upper bound on block sensitivity and certificate complexity in terms of sensitivity
摘要：
灵敏度\cite{CD82,CDR86}和块灵敏度\cite{Nisan91}是布尔函数复杂性的两个重要度量。决策树复杂性中的一个长期存在的开放问题，即由Nisan和Szegedy \cite{Nisan94}在1992年提出的“灵敏度与块灵敏度”问题，是这两个复杂性度量是否多项式相关，即，是否为$bs(f)=O(s(f)^{O(1)})$。我们证明了一个新的上界块灵敏度的灵敏度：$bs(f) \leq 2^{s(f)-1} s(f)$。以前，块灵敏度的最佳上限是Kenyon和Kendy \cite{KK}的$bs(f) \leq (\frac{e}{\sqrt{2π}}) e^{s(f)} \sqrt{s(f)}$。我们还证明，如果$\min\{s_0(f),s_1(f)\}$是一个常数，那么灵敏度和块灵敏度是线性相关的，即$bs(f)=O(s(f))$。
英文摘要：
Sensitivity \cite{CD82,CDR86} and block sensitivity \cite{Nisan91} are two important complexity measures of Boolean functions. A longstanding open problem in decision tree complexity, the "Sensitivity versus Block Sensitivity" question, proposed by Nisan and Szegedy \cite{Nisan94} in 1992, is whether these two complexity measures are polynomially related, i.e., whether $bs(f)=O(s(f)^{O(1)})$. We prove an new upper bound on block sensitivity in terms of sensitivity: $bs(f) \leq 2^{s(f)-1} s(f)$. Previously, the best upper bound on block sensitivity was $bs(f) \leq (\frac{e}{\sqrt{2π}}) e^{s(f)} \sqrt{s(f)}$ by Kenyon and Kutin \cite{KK}. We also prove that if $\min\{s_0(f),s_1(f)\}$ is a constant, then sensitivity and block sensitivity are linearly related, i.e. $bs(f)=O(s(f))$.
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