We consider the task of detecting a hidden bipartite subgraph in a given random graph. Specifically, under the null hypothesis, the graph is a realization of an Erd\H{o}s-R\'{e}nyi random graph over $n$ vertices with edge density $q$. Under the alternative, there exists a planted $k_{\mathsf{R}} \times k_{\mathsf{L}}$ bipartite subgraph with edge density $p>q$. We derive asymptotically tight upper and lower bounds for this detection problem in both the dense regime, where $q,p = \Theta\left(1\right)$, and the sparse regime where $q,p = \Theta\left(n^{-\alpha}\right), \alpha \in \left(0,2\right]$. Moreover, we consider a variant of the above problem, where one can only observe a relatively small part of the graph, by using at most $\mathsf{Q}$ edge queries. For this problem, we derive upper and lower bounds in both the dense and sparse regimes.

翻译：我们考虑在给定随机图中检测隐藏二分子图的任务。具体地，在原假设下，该图是边密度为$q$的$n$个顶点上的Erdős–Rényi随机图的一个实现。在备择假设下，存在一个边密度为$p>q$的植入$k_{\mathsf{R}} \times k_{\mathsf{L}}$二分子图。我们推导了该检测问题在稠密情形（即$q,p = \Theta\left(1\right)$）和稀疏情形（即$q,p = \Theta\left(n^{-\alpha}\right), \alpha \in \left(0,2\right]$）下的渐近紧的上下界。此外，我们考虑了上述问题的一个变体，其中通过最多使用$\mathsf{Q}$次边查询，只能观测到图的相对较小部分。针对该问题，我们推导了稠密和稀疏情形下的上下界。