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

The Algorithmic Phase Transition of Random Graph Alignment Problem