We study the graph alignment problem over two independent Erd\H{o}s-R\'enyi graphs on $n$ vertices, with edge density $p$ falling into two regimes separated by the critical window around $p_c=\sqrt{\log n/n}$. Our result reveals an algorithmic phase transition for this random optimization problem: polynomial-time approximation schemes exist in the sparse regime, while statistical-computational gap emerges in the dense regime. Additionally, we establish a sharp transition on the performance of online algorithms for this problem when $p$ lies in the dense regime, resulting in a $\sqrt{8/9}$ multiplicative constant factor gap between achievable and optimal solutions.

翻译：我们研究两个独立的$n$顶点Erdős–Rényi随机图上的图对齐问题，其边密度$p$落入由临界窗口$p_c=\sqrt{\log n/n}$分隔的两个区间。我们的结果揭示了该随机优化问题的一个算法相变：在稀疏区间存在多项式时间近似方案，而在稠密区间则出现统计计算差距。此外，当$p$处于稠密区间时，我们建立了该问题在线算法性能的尖锐转变，导致可达解与最优解之间存在$\sqrt{8/9}$的乘法常数因子差距。