arXiv:2604.13577 · v2

Lower Bounds for Testing Directed Acyclicity in the Unidirectional Bounded-Degree Model

首次提交 2026-04-15 15:38

版本更新 2026-11-12 10:37

雷达观测 2026-11-13 14:16

摘要

We study property testing of directed acyclicity in the unidirectional bounded-degree oracle model, where a query to a vertex reveals its outgoing neighbors. We prove that there exist absolute constants $d_0\in\mathbb{N}$ and $\varepsilon>0$ such that for every constant $d\ge d_0$, any one-sided $\varepsilon$-tester for acyclicity on $n$-vertex digraphs of maximum outdegree at most $d$ requires $\widetildeΩ(n^{2/3})$ queries. This improves the previous $\widetildeΩ(n^{5/9})$ lower bound for one-sided testing of acyclicity in the same model. We also prove that, under the same degree assumption, any two-sided $\varepsilon$-tester requires $Ω(\sqrt n)$ queries, improving the previous $Ω(n^{1/3})$ lower bound. We further prove an $Ω(n)$ lower bound for tolerant testing for some absolute constant outdegree bound $d$ by reduction from bounded-degree $3$-colorability.

作者备注：The author has withdrawn this paper due to an error in the proof of the one-sided lower bound. In particular, the bound on the probability of a long blue path in Lemma 4.3 does not hold for adaptive exploration.