The statistical threshold for planted matchings and spanning trees

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.