Detection of a planted dense subgraph in a random graph is a fundamental statistical and computational problem that has been extensively studied in recent years. We study a hypergraph version of the problem. Let $G^r(n,p)$ denote the $r$-uniform Erd\H{o}s-R\'enyi hypergraph model with $n$ vertices and edge density $p$. We consider detecting the presence of a planted $G^r(n^\gamma, n^{-\alpha})$ subhypergraph in a $G^r(n, n^{-\beta})$ hypergraph, where $0< \alpha < \beta < r-1$ and $0 < \gamma < 1$. Focusing on tests that are degree-$n^{o(1)}$ polynomials of the entries of the adjacency tensor, we determine the threshold between the easy and hard regimes for the detection problem. More precisely, for $0 < \gamma < 1/2$, the threshold is given by $\alpha = \beta \gamma$, and for $1/2 \le \gamma < 1$, the threshold is given by $\alpha = \beta/2 + r(\gamma - 1/2)$. Our results are already new in the graph case $r=2$, as we consider the subtle log-density regime where hardness based on average-case reductions is not known. Our proof of low-degree hardness is based on a conditional variant of the standard low-degree likelihood calculation.

翻译：在随机图中检测植入的稠密子图是一个基础的统计与计算问题，近年来受到广泛研究。本文研究该问题的超图版本。设$G^r(n,p)$表示具有$n$个顶点和边密度$p$的$r$元Erdős–Rényi超图模型。我们考虑在$G^r(n, n^{-\beta})$超图中检测是否存在植入的$G^r(n^\gamma, n^{-\alpha})$子超图，其中$0< \alpha < \beta < r-1$且$0 < \gamma < 1$。聚焦于以邻接张量条目的$n^{o(1)}$次多项式为检验函数，我们确定了检测问题中易解与困难机制之间的阈值。具体而言，当$0 < \gamma < 1/2$时，阈值为$\alpha = \beta \gamma$；当$1/2 \le \gamma < 1$时，阈值为$\alpha = \beta/2 + r(\gamma - 1/2)$。即使在图情形$r=2$下，我们的结果也具有新意，因为研究涉及基于平均情况归约无法证明困难性的微妙对数密度区域。低度困难性的证明基于标准低度似然计算的条件变体。