Testing Monotonicity of Real-Valued Functions on DAGs

TL;DR

The paper advances the theory of monotonicity testing for real-valued functions on explicit DAGs by establishing near-optimal lower bounds and sharper upper bounds. It introduces positive-matching RS (PMRS) families to translate RS-style lower-bound templates to the real-valued setting, enabling $ ext{Ω}(n^{1/2- ext{δ}}/ ext{√ε})$ non-adaptive two-sided lower bounds and $ ext{Ω}( ext{√}n)$ adaptive one-sided lower bounds via softly planted violations and careful information leakage control. On the algorithmic side, it provides non-adaptive one-sided testers with improved dependence on the transitive reduction and closure, achieving $O( ext{√}{m ext{ℓ}}/( ext{ε}n))$ and $O(m^{1/3}/ ext{ε}^{2/3})$ query bounds, which are advantageous in sparse regimes where $m ext{ℓ}=o(n^3)$ or $m=o(n^{3/2})$. The work combines combinatorial constructions (PMRS), probabilistic/analytic tools (Gibbs measures, Green-function comparisons), and graph-theoretic routing arguments to bridge lower and upper bounds, yielding a cohesive understanding of monotonicity testing on DAGs with real-valued ranges and its practical implications for sublinear testing in structured domains.

Abstract

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.

Figures (1)

• Figure 1: Comparison of query complexities for monotonicity testing, where $m=\Theta(n^c)$ and $\ell=\Theta(n^d)$ and $\epsilon>0$ is a constant. If the underlying undirected graph is connected, then $n\le m\le n^2$ so $1\le c\le 2$, and since $m\le \ell\le n^2$ we have $c\le d\le 2$. Each colored region indicates which query complexity achieves the lowest value. Red: $O(\sqrt{m\ell}/(\varepsilon n))$ (\ref{['lem:sqrt-ml']}), Blue: $O(m^{1/3}/\varepsilon^{2/3})$ (\ref{['lem:m13']}), Yellow: $O(\sqrt{n/\varepsilon})$ (fischer2002monotonicity). The additional bound $O(\ell/(\varepsilon n))$ is omitted since it is never optimal in this range.

Theorems & Definitions (84)

• Definition 1: Positive matching farrokhi_gharakhloo_yazdanpour_pmd_2021
• Definition 2: Alternating walks and cycles
• Theorem 2.1: Characterization of positive matchings farrokhi_gharakhloo_yazdanpour_pmd_2021golumbic_hirst_lewenstein_2001
• Lemma 1: Violations exactly on a matching
• proof
• Lemma 2: Submatchings preserve positivity
• proof
• Definition 3: PMRS: Positive-Matching RS family
• Lemma 3: A violated matching forces $\varepsilon$-farness
• proof