The statistical threshold for planted matchings and spanning trees

TL;DR

This paper investigates the detectability of planted matchings and spanning trees in Erdős–Rényi graphs, establishing thresholds for when such structures can be reliably identified using efficient algorithms.

Contribution

It determines the statistical thresholds for detecting planted structures in random graphs and shows the computational feasibility of detection below these thresholds.

Findings

Detection is impossible when q ≫ n^{-1/2}.

Efficient tests succeed when q ≪ n^{-1/2}.

Threshold at q ≈ n^{-1/2} for detection capability.

Abstract

In this paper, we study the problem of detecting the presence of a planted perfect matching or spanning tree in an Erd\H{o}s--R\'enyi 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 Erd\H{o}s--R\'enyi random graph $G(n,q)$, while under the alternative hypothesis, the graph is the union of an Erd\H{o}s--R\'enyi 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.