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：我们提出一个有效的图表匹配算法, 其依据是计算每个顶端根根根根的某一组加权树的相近分数。 对于两张Erd\H{o}s- R\'enyi polices $\mathcal{G}(n,q) 美元, 其边际通过潜在的顶端对应而相关联的两张Erd\H}H{o}s- R\'enyi 图形$\ mathcal{G}(n, q) 美元, 我们表明, 这个算法正确匹配了所有顶端与高概率的一个消失部分, 但它是一个非常稳定的匹配算法, 并且适用于稀薄和稠密的图形。 相比之下, 之前的方法要么需要$\ró=2 ⁇ alpha\ apprrox 0. 338$, $\ approgrox0. 338$, 美元是 Opprox 0. 338$, $\\ dalpha$ 是 Orter's the transcocountial countings。 rquest rquest the crequest the cregregregrequest