SIAM Journal on Computing, Ahead of Print.
We show that every algorithm for testing $n$-variate Boolean functions for monotonicity must have query complexity $\tilde{\Omega}(n^{1/4})$. All previous lower bounds for this problem were designed for nonadaptive algorithms and, as a result, the best previous lower bound for general (possibly adaptive) monotonicity testers was only $\Omega(\log n)$. Combined with the query complexity of the nonadaptive monotonicity tester of Khot, Minzer, and Safra (FOCS 2015), our lower bound shows that adaptivity can result in at most a quadratic reduction in the query complexity for testing monotonicity. By contrast, we show that there is an exponential gap between the query complexity of adaptive and nonadaptive algorithms for testing regular linear threshold functions (LTFs) for monotonicity. Chen, De, Servedio, and Tan (STOC 2015) recently showed that nonadaptive algorithms require almost $\Omega(n^{1/2})$ queries for this task. We introduce a new adaptive monotonicity testing algorithm which has query complexity $O(\log n)$ when the input is a regular LTF.

中文翻译：

---

用于测试单调性的多项式下界

SIAM 计算杂志，超前印刷。
我们表明，每个用于测试 $n$-variate 布尔函数单调性的算法都必须具有查询复杂度 $\tilde{\Omega}(n^{1/4})$。这个问题之前的所有下界都是为非自适应算法设计的，因此，一般（可能是自适应的）单调性测试者的最佳先前下界仅为 $\Omega(\log n)$。结合 Khot、Minzer 和 Safra（FOCS 2015）的非自适应单调性测试器的查询复杂性，我们的下限表明自适应性最多可以导致测试单调性的查询复杂性的二次降低。相比之下，我们表明，用于测试常规线性阈值函数 (LTF) 的单调性的自适应和非自适应算法的查询复杂性之间存在指数差距。陈德，Servedio，和 Tan (STOC 2015) 最近表明，非自适应算法几乎需要 $\Omega(n^{1/2})$ 查询来完成这项任务。我们引入了一种新的自适应单调性测试算法，当输入是常规 LTF 时，该算法具有查询复杂度 $O(\log n)$。