We study monotonicity testing of real-valued functions on directed acyclic graphs (DAGs) with $n$ vertices. For every constant $δ>0$, we prove a $Ω(n^{1/2-δ}/\sqrt{\varepsilon})$ lower bound against non-adaptive two-sided testers on DAGs, nearly matching the classical $O(\sqrt{n/\varepsilon})$-query upper bound. For constant $\varepsilon$, we also prove an $Ω(\sqrt n)$ lower bound for randomized adaptive one-sided testers on explicit bipartite DAGs, whereas previously only an $Ω(\log n)$ lower bound was known. A key technical ingredient in both lower bounds is positive-matching Ruzsa--Szemerédi families. On the algorithmic side, we give simple non-adaptive one-sided testers with query complexity $O(\sqrt{m\,\ell}/(\varepsilon n))$ and $O(m^{1/3}/\varepsilon^{2/3})$, where $m$ is the number of edges in the transitive reduction and $\ell$ is the number of edges in the transitive closure. For constant $\varepsilon>0$, these improve over the previous $O(\sqrt{n/\varepsilon})$ bound when $m\ell=o(n^3)$ and $m=o(n^{3/2})$, respectively.

翻译：我们研究在具有 $n$ 个顶点的有向无环图（DAG）上实值函数的单调性测试问题。对于任意常数 $δ>0$，我们证明了在 DAG 上非自适应双边测试器的 $Ω(n^{1/2-δ}/\sqrt{\varepsilon})$ 下界，这几乎匹配了经典的 $O(\sqrt{n/\varepsilon})$ 查询上界。对于常数 $\varepsilon$，我们还在显式构造的二部 DAG 上证明了随机自适应单边测试器的 $Ω(\sqrt n)$ 下界，而此前仅知 $Ω(\log n)$ 下界。这两个下界证明的关键技术工具是正匹配 Ruzsa--Szemerédi 族。在算法方面，我们给出了简单的非自适应单边测试器，其查询复杂度分别为 $O(\sqrt{m\,\ell}/(\varepsilon n))$ 和 $O(m^{1/3}/\varepsilon^{2/3})$，其中 $m$ 是传递归约中的边数，$\ell$ 是传递闭包中的边数。对于常数 $\varepsilon>0$，当分别满足 $m\ell=o(n^3)$ 和 $m=o(n^{3/2})$ 时，这些结果改进了先前 $O(\sqrt{n/\varepsilon})$ 的界。