Planted Bipartite Graph Detection

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 Erdos-Rényi random graph over n vertices with edge density q. Under the alternative, there exists a planted k_R × k_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 = ⊝ (1), and the sparse regime where p, q = ⊝ (n^-α), α ⋯ (0, 2]. 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.