The statistical threshold for planted matchings and spanning trees

TL;DR

This work analyzes the detectability of planted subgraphs—specifically a perfect matching and a spanning tree—within Erdős--Rényi graphs after adjusting the null to offset edge-count differences. By deriving and exploiting the likelihood-ratio framework, the authors reduce detection to controlling the chi-squared distance via higher-order moments tied to collisions among planted subgraphs, and then show a sharp threshold at $p=\Theta(n^{-1/2})$ (equivalently, at $q$ relative to the planted structure) separating impossible from possible detection. In the sparse regime $p=o(n^{-1/2})$, simple computationally efficient tests succeed with high probability, while for $p=\omega(n^{-1/2})$ no test can outperform random guessing, indicating no statistical–computational gap in these settings. The results rely on a Poisson-collision approximation for the planted objects and, in the spanning-tree case, on negative association of tree edges to bound the likelihood ratio, highlighting a nuanced boundary where sparse planted structures become detectable despite edge-count adjustments.

Abstract

In this paper, we study the problem of detecting the presence of a planted perfect matching or spanning tree in an Erdős--Rényi random graph. More precisely, we study the hypothesis testing problem where the statistician observes a graph on $n$ vertices. Under the null hypothesis, the graph is a realization of an Erdős--Rényi random graph $G(n,q)$, while under the alternative hypothesis, the graph is the union of an Erdős--Rényi random graph and a random perfect matching (or random spanning tree). In order to avoid trivial detection by counting edges, we adjust the alternative hypothesis so that the expected number of edges under both distributions coincides. We prove that in both problems, when $q\gg n^{-1/2}$, no test can perform better than random guessing, while for $q\ll n^{-1/2}$, there exist computationally efficient tests that guess correctly with high probability.

Theorems & Definitions (12)

• Theorem 3.1
• Lemma 3.2
• proof
• Claim 3.3
• proof
• Lemma 3.4
• proof : Proof of \ref{['lem:test-match']}
• Theorem 4.1
• Lemma 4.3
• proof