超越Talagrand函数：检验单调性和不单调性的新下界
Beyond Talagrand Functions: New Lower Bounds for Testing Monotonicity and Unateness
摘要：
我们证明了一个下界$\tildeΩ(n^{1/3})$的查询复杂性的任何双边和自适应算法，测试是否一个未知的布尔函数$f:\{0,1\}^n\rightarrow \{0,1\}$是单调或远离单调。这改进了Belovs和Blais [BB 15]针对同一问题的$\tildeΩ(n^{1/4})$的最近界。我们的结果建立在一个新的家庭的随机布尔函数，可以被看作是一个两级扩展的Talagrand的随机DNF。除了单调性，我们还证明了一个下界的$\tildeΩ(n^{2/3})$的任何双边和自适应算法，和一个下界的$\tildeΩ(n)$的任何单边和非自适应算法的测试unconstant，单调性的自然推广。后者与Khot和Shinkar [KS 15]以及Chakrabarty和Seshadhri [CS 16]最近的线性上界相匹配。
英文摘要：
We prove a lower bound of $\tildeΩ(n^{1/3})$ for the query complexity of any two-sided and adaptive algorithm that tests whether an unknown Boolean function $f:\{0,1\}^n\rightarrow \{0,1\}$ is monotone or far from monotone. This improves the recent bound of $\tildeΩ(n^{1/4})$ for the same problem by Belovs and Blais [BB15]. Our result builds on a new family of random Boolean functions that can be viewed as a two-level extension of Talagrand's random DNFs. Beyond monotonicity, we also prove a lower bound of $\tildeΩ(n^{2/3})$ for any two-sided and adaptive algorithm, and a lower bound of $\tildeΩ(n)$ for any one-sided and non-adaptive algorithm for testing unateness, a natural generalization of monotonicity. The latter matches the recent linear upper bounds by Khot and Shinkar [KS15] and by Chakrabarty and Seshadhri [CS16].
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