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 $n$ vertices. 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 relatively small part of the graph using adaptive edge queries. For this model, we determine the number of queries necessary and sufficient (accompanied with a quasi-polynomial optimal algorithm) for detecting the presence of the planted subgraph. We also propose a polynomial-time algorithm which is able to detect the planted subgraph, albeit with more queries compared to the above lower bound. We conjecture that in the leftover regime, no polynomial-time algorithms exist. Our results resolve two open questions posed in the past literature.

翻译：种植稠密子图检测问题指的是检验给定的（随机）图中是否存在异常稠密的子图。具体而言，我们观测一个包含n个顶点的无向无权图。在原假设下，该图是边概率（或密度）为q的Erdős–Rényi图的一个实现。在备择假设下，存在一个包含k个顶点的子图，其边概率为p>q。该问题的统计和计算障碍在边参数p和q的广泛取值范围内已得到充分理解。在本文中，我们考虑上述问题的一个自然变体，即只能通过自适应边查询观测图的相对较小部分。针对该模型，我们确定了检测种植子图存在所需且充分的查询次数（伴随一个拟多项式时间的最优算法）。我们还提出了一种多项式时间算法，尽管所需的查询次数高于上述下界，但仍能检测到种植子图。我们推测在剩余参数范围内，不存在多项式时间算法。我们的结果解决了过往文献中提出的两个开放问题。