We propose an efficient algorithm for matching two correlated Erd\H{o}s--R\'enyi graphs with $n$ vertices whose edges are correlated through a latent vertex correspondence. When the edge density $q= n^{- \alpha+o(1)}$ for a constant $\alpha \in [0,1)$, we show that our algorithm has polynomial running time and succeeds to recover the latent matching as long as the edge correlation is non-vanishing. This is closely related to our previous work on a polynomial-time algorithm that matches two Gaussian Wigner matrices with non-vanishing correlation, and provides the first polynomial-time random graph matching algorithm (regardless of the regime of $q$) when the edge correlation is below the square root of the Otter's constant (which is $\approx 0.338$).

翻译：我们提出了一种高效算法，用于匹配两个具有$n$个顶点的相关Erdős–Rényi图，这些图的边通过潜在顶点对应关系相关联。当边密度$q= n^{-\alpha+o(1)}$（其中常数$\alpha \in [0,1)$）时，我们证明该算法具有多项式运行时间，并能在边相关性非零的情况下成功恢复潜在匹配。这与我们此前关于匹配两个非零相关高斯Wigner矩阵的多项式时间算法的工作密切相关，并且首次提供了当边相关性低于Otter常数平方根（约为0.338）时的多项式时间随机图匹配算法（无论$q$处于何种区域）。