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

2026-08-06

AI Summary

This study investigates whether low-degree heuristics can rigorously establish computational indistinguishability in noisy planted subgraph detection. Focusing on the model where a deterministic subgraph $\Gamma_n$ is planted into an Erdős–Rényi random graph $G(n, c/n)$, the authors integrate Fourier analysis, treewidth decomposition, and subgraph counting techniques to incorporate, for the first time within the low-degree framework, both automorphism factors and counts of isomorphic triples. They prove that the planted model is asymptotically indistinguishable from the null model when either $c \leq 1$ and $D_n = \omega(\log n)$, or $c > 1$ and $\mathrm{tw}(\Gamma_n) = o(D_n / \log n)$. This work provides a precise characterization of the theoretical limits of low-degree methods in sparse graph detection, introducing a treewidth condition in the supercritical regime while showing it is unnecessary in the subcritical and critical regimes.

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.

Problem

• planted subgraph detection
• low-degree heuristic
• noise tolerance
• treewidth
• computational thresholds

Innovation

• low-degree method
• planted subgraph detection
• treewidth
• noise robustness
• Fourier analysis on graphs