Recovery of Planted Subgraphs

Understanding the fundamental limits of recovering planted subgraphs in random graphs is a central challenge in high-dimensional statistics and theoretical computer science. While existing work has largely focused on special subgraph families such as cliques, bicliques, or dense blocks, the exact recovery of a general planted subgraph in Erdős–Rényi random graphs remains poorly understood. In this paper, we study the exact recovery of an arbitrary planted subgraph $\Gamma = \Gamma_n$ embedded in a dense Erdős–Rényi random graph $\mathcal{G}(n,q_n)$, where edges within $\Gamma$ are present independently with probability $p_n > q_n$. Our main results identify sharp conditions under which exact recovery is possible with high probability, and we establish matching lower bounds showing the necessity of these conditions. The resulting statistical threshold is characterized by a new graph-theoretic quantity, which we term the \emph{minimal maximum subgraph density}. This quantity is defined as the maximum subgraph density of the smallest induced balanced subgraph of $\Gamma$. We then turn to the problem of recovery under polynomial-time constraints. We propose a computationally efficient recovery algorithm that applies to arbitrary planted subgraphs and analyze its performance in terms of certain spectral properties of the adjacency matrix. In addition, we derive computational lower bounds for recovery using the low-degree polynomial framework, establishing regimes where recovery is statistically possible but computationally hard. Finally, we consider several extensions of our setting, including recovery in semi-random models and weaker notions of recovery.