Testing Monotonicity of Real-Valued Functions on DAGs

We study monotonicity testing of real-valued functions on directed acyclic graphs (DAGs) with $n$ vertices. For every constant $\delta>0$, we prove a $\Omega(n^{1/2-\delta}/\sqrt{\varepsilon})$ 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 $\Omega(\sqrt n)$ lower bound for randomized adaptive one-sided testers on explicit bipartite DAGs, whereas previously only an $\Omega(\log n)$ lower bound was known. A key technical ingredient in both lower bounds is positive-matching Ruzsa--Szemer\'edi 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.