Random Subgraph Detection Using Queries

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 finds the planted subgraph with high probability by making $\mathsf{Q} = O(\frac{n^2}{k^2\chi^4(p||q)}\log^2n)$ adaptive queries. We then propose a polynomial-time algorithm which is able to detect the planted subgraph using $\mathsf{Q} = O(\frac{n^4}{k^4\chi^2(p||q)}\log n)$ queries. We conjecture that in the leftover regime, where $\frac{n^2}{k^2}\ll\mathsf{Q}\ll \frac{n^4}{k^4}$, no polynomial-time algorithms exist; we give an evidence for this hypothesis using the planted clique conjecture. Our results resolve three questions posed in \cite{racz2020finding}, where the special case of adaptive detection and recovery of a planted clique was considered.