Rigorous Low-Degree Implications for Planted Subgraph Detection: Noise and Treewidth

Abstract

The low-degree heuristic has become a widely used framework for predicting computational thresholds in average-case planted-versus-null problems. However, a recent sequence of counterexamples shows that low-degree indistinguishability does not, in general, rule out efficient noise-tolerant distinguishers; see Buhai et al. (2025) and Mao (2026). Motivated by these developments, Hsieh et al. (2026) initiated the study of rigorous consequences of the low-degree heuristic. In this work, we continue this program for planted-graph problems. Let $Q_n=G(n,c/n)$, and let $P_n$ be obtained by planting a uniformly random copy of a deterministic graph $Γ_n$ into an independent sample from $Q_n$. In the supercritical regime $c>1$, we show that if $P_n$ is degree-$D_n$ indistinguishable from $Q_n$ and $\operatorname{tw}(Γ_n)=o(D_n/\log n)$, then a noisy version of $P_n$ is asymptotically indistinguishable from $Q_n$. Here $\operatorname{tw}(Γ_n)$ denotes the treewidth of $Γ_n$, a measure of how efficiently the graph can be decomposed into tree-like pieces. In the critical and subcritical regimes $0<c\leq 1$, the same conclusion holds whenever $D_n=ω(\log n)$, without any treewidth assumption. Our proof has two main ingredients. First, we uncover a correspondence between the subgraph-count and automorphism factors in the Fourier expansion and counts of isomorphism triples. Second, we cut the decomposition tree into subtrees, breaking each large Fourier support into low-degree pieces that meet at only a few interface vertices, and use noise to absorb the cost of reassembling them. At and below criticality, the low-degree assumption rules out short cycles, while noise destroys the remaining long cycles.