  We consider the task of detecting a hidden bipartite subgraph in a given
random graph. This is formulated as a hypothesis testing problem, 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 characterize the statistical and computational barriers
for this problem. Specifically, we derive information-theoretic lower bounds,
and design and analyze optimal algorithms matching those bounds, in both the
dense regime, where $p,q = \Theta\left(1\right)$, and the sparse regime where
$p,q = \Theta\left(n^{-\alpha}\right), \alpha \in \left(0,2\right]$. We also
consider the problem of testing in polynomial-time. As is customary in similar
structured high-dimensional problems, our model undergoes an
"easy-hard-impossible" phase transition and computational constraints penalize
the statistical performance. To provide an evidence for this statistical
computational gap, we prove computational lower bounds based on the low-degree
conjecture, and show that the class of low-degree polynomials algorithms fail
in the conjecturally hard region.
