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.

翻译：图对齐问题旨在识别两个相关图之间的顶点对应关系。现有研究大多集中于两个图共享相同顶点集的场景。然而，在许多实际应用中，例如计算机视觉、社交网络分析和生物信息学，任务通常涉及在较大图中定位一个较小的图模式。现有的图对齐算法和分析无法直接处理这些场景，因为它们并非设计用于识别较小图模式在较大图中所处的特定顶点子集。受此局限性的启发，我们提出了子图对齐问题，该问题旨在同时恢复嵌入较大图中的较小图模式的顶点集和/或顶点对应关系。在较小图模式是较大图的诱导子图且需要同时恢复顶点集与对应关系的特殊情况下，该问题可简化为子图同构问题，而后者在最坏情况下是NP完全的。本文通过提出Erdos-Renyi子图对模型及适当的恢复准则，形式化地构建了子图对齐问题。随后，我们为该问题建立了几乎紧的信息论结果，并提出了一些新颖的分析方法。