We propose an efficient algorithm for graph matching based on similarity scores constructed from counting a certain family of weighted trees rooted at each vertex. For two Erd\H{o}s-R\'enyi graphs $\mathcal{G}(n,q)$ whose edges are correlated through a latent vertex correspondence, we show that this algorithm correctly matches all but a vanishing fraction of the vertices with high probability, provided that $nq\to\infty$ and the edge correlation coefficient $\rho$ satisfies $\rho^2>\alpha \approx 0.338$, where $\alpha$ is Otter's tree-counting constant. Moreover, this almost exact matching can be made exact under an extra condition that is information-theoretically necessary. This is the first polynomial-time graph matching algorithm that succeeds at an explicit constant correlation and applies to both sparse and dense graphs. In comparison, previous methods either require $\rho=1-o(1)$ or are restricted to sparse graphs. The crux of the algorithm is a carefully curated family of rooted trees called chandeliers, which allows effective extraction of the graph correlation from the counts of the same tree while suppressing the undesirable correlation between those of different trees.

翻译：我们提出了一种基于加权有根树族计数的相似性得分的图匹配高效算法。对于通过潜在顶点对应关系相关的两个Erdős–Rényi图\(\mathcal{G}(n,q)\)，我们证明：当\(nq\to\infty\)且边相关系数\(\rho\)满足\(\rho^2>\alpha \approx 0.338\)（其中\(\alpha\)为Otter树计数常数）时，该算法能以高概率正确匹配除可忽略比例顶点外的所有顶点。此外，在信息论必要的额外条件下，这种近乎精确的匹配可达到完全精确。这是首个在显式常数相关度下成功运行的多项式时间图匹配算法，同时适用于稀疏和稠密图。相比之下，先前的方法要么要求\(\rho=1-o(1)\)，要么仅限于稀疏图。算法的关键在于精心设计的称为"枝形烛台"的有根树族，它能在抑制不同树计数之间不良相关性的同时，有效从相同树的计数中提取图的相关性。