planted 匹配和生成树的统计阈值
The statistical threshold for planted matchings and spanning trees
AI总结：
研究了在 Erdős--Rényi 随机图中检测 planted 完美匹配或生成树的统计阈值问题，证明了在不同 q 值下检测的可行性。
摘要：
在本文中，我们研究了在 Erdős--Rényi 随机图中检测 planted 完美匹配或生成树存在的问题。更具体地说，我们研究了假设检验问题，统计学家观察一个有 n 个顶点的图。在原假设下，图是 Erdős--Rényi 随机图 G(n,q) 的实现，而在备择假设下，图是 Erdős--Rényi 随机图和随机完美匹配（或随机生成树）的并集。为了避免通过计数边进行简单检测，我们将备择假设调整为两种分布下的边数期望相同。我们证明在两个问题中，当 q ≫ n^{-1/2} 时，没有测试能比随机猜测更好，而当 q ≪ n^{-1/2} 时，存在计算上高效的测试，可以以高概率正确猜测。
英文摘要：
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.
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