The planted densest subgraph detection problem refers to the task of testing whether in a given (random) graph there is a subgraph that is unusually dense. Specifically, we observe an undirected and unweighted graph on $n$ nodes. Under the null hypothesis, the graph is a realization of an Erd\H{o}s-R\'{e}nyi graph with edge probability (or, density) $q$. Under the alternative, there is a subgraph on $k$ vertices with edge probability $p>q$. The statistical as well as the computational barriers of this problem are well-understood for a wide range of the edge parameters $p$ and $q$. In this paper, we consider a natural variant of the above problem, where one can only observe a small part of the graph using adaptive edge queries. For this model, we determine the number of queries necessary and sufficient for detecting the presence of the planted subgraph. Specifically, we show that any (possibly randomized) algorithm must make $\mathsf{Q} = \Omega(\frac{n^2}{k^2\chi^4(p||q)}\log^2n)$ adaptive queries (on expectation) to the adjacency matrix of the graph to detect the planted subgraph with probability more than $1/2$, where $\chi^2(p||q)$ is the Chi-Square distance. On the other hand, we devise a quasi-polynomial-time algorithm that detects the planted subgraph with high probability by making $\mathsf{Q} = O(\frac{n^2}{k^2\chi^4(p||q)}\log^2n)$ non-adaptive queries. We then propose a polynomial-time algorithm which is able to detect the planted subgraph using $\mathsf{Q} = O(\frac{n^3}{k^3\chi^2(p||q)}\log^3 n)$ queries. We conjecture that in the leftover regime, where $\frac{n^2}{k^2}\ll\mathsf{Q}\ll \frac{n^3}{k^3}$, no polynomial-time algorithms exist. Our results resolve two questions posed in \cite{racz2020finding}, where the special case of adaptive detection and recovery of a planted clique was considered.

翻译：安装密度最深的子图检测问题是指在给定的 {( random) 图形中测试$2 的值是否为异常稠密。 具体地说, 我们观察的是美元节点上的未方向和未加权的图表。 在无效假设下, 该图是一个带有边缘概率( 或, 密度) 的 Erd\ H{ { { e} 尼基图的实现。 在另一个选项下, 有一份有关 $k$ 的垂直值的子图, 其边缘概率为 $2 美元 。 这一问题的统计和计算屏障非常深。 在本文中, 我们考虑的是上述问题的自然变量, 其中只能使用适应性边缘查询来观察小部分 。 对于这个模型, 我们确定必要的查询次数, 并足以检测到配置子图的存在 。 具体地, 我们显示任何( 可能随机的) 算的算法都必须使用 $math= 美元( ) com- ción2 的直径( ción) 。