Information-Theoretic Limits on Exact Subgraph Alignment Problem

The graph alignment problem aims to identify the vertex correspondence between two correlated graphs. Most existing studies focus on the scenario in which the two graphs share the same vertex set. However, in many real-world applications, such as computer vision, social network analysis, and bioinformatics, the task often involves locating a small graph pattern within a larger graph. Existing graph alignment algorithms and analysis cannot directly address these scenarios because they are not designed to identify the specific subset of vertices where the small graph pattern resides within the larger graph. Motivated by this limitation, we introduce the subgraph alignment problem, which seeks to recover both the vertex set and/or the vertex correspondence of a small graph pattern embedded in a larger graph. In the special case where the small graph pattern is an induced subgraph of the larger graph and both the vertex set and correspondence are to be recovered, the problem reduces to the subgraph isomorphism problem, which is NP-complete in the worst case. In this paper, we formally formulate the subgraph alignment problem by proposing the Erdos-Renyi subgraph pair model together with some appropriate recovery criterion. We then establish almost-tight information-theoretic results for the subgraph alignment problem and present some novel approaches for the analysis.