The Algorithmic Phase Transition of Random Graph Alignment Problem

TL;DR

This paper investigates the graph alignment problem on Erdős-Rényi graphs, revealing an algorithmic phase transition at a critical edge density where polynomial algorithms succeed in sparse regimes but fail in dense regimes due to a statistical-computational gap.

Contribution

It identifies a sharp phase transition in the algorithmic complexity of graph alignment around a critical edge density, and characterizes the performance gap of online algorithms in dense regimes.

Findings

Polynomial-time approximation schemes exist in the sparse regime.

A statistical-computational gap appears in the dense regime.

Online algorithms have a performance gap with a actor in dense regimes.

Abstract

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.