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The Algorithmic Phase Transition of Random Graph Alignment Problem

Hang Du, Shuyang Gong, Rundong Huang

TL;DR

The work analyzes the graph alignment problem on two independent ER graphs, uncovering an informational and computational phase structure across a sparse-to-dense transition near $p_c=\sqrt{\log n/n}$. It proves that in the sparse regime, polynomial-time algorithms can achieve near-optimal alignment (PTAS) while in the dense regime a statistical–computational gap emerges, evidenced by an overlap-gap phenomenon and hard online settings. A simple greedy framework $\mathcal{A}_\eta$ achieves near-optimal performance in both regimes, delivering a sharp algorithmic threshold at $\beta_c=\sqrt{8/9}$ for online methods. The combination of first/second moment analyses, Talagrand concentration, and sophisticated OGP-based hardness arguments provides a comprehensive picture of when efficient algorithms can succeed and when inherent computational barriers arise for random GAP instances. Key contributions include (i) precise informational thresholds in sparse and dense regimes, (ii) a constructive greedy algorithm achieving near-optimal overlaps in both regimes, (iii) a rigorous 2-OGP-based barrier for stable algorithms, and (iv) a branching-OGP-based hardness result for online algorithms establishing the threshold $\beta_c$ as a fundamental computational limit in the dense regime.

Abstract

We study the graph alignment problem over two independent Erdős-Rényi 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.

The Algorithmic Phase Transition of Random Graph Alignment Problem

TL;DR

The work analyzes the graph alignment problem on two independent ER graphs, uncovering an informational and computational phase structure across a sparse-to-dense transition near . It proves that in the sparse regime, polynomial-time algorithms can achieve near-optimal alignment (PTAS) while in the dense regime a statistical–computational gap emerges, evidenced by an overlap-gap phenomenon and hard online settings. A simple greedy framework achieves near-optimal performance in both regimes, delivering a sharp algorithmic threshold at for online methods. The combination of first/second moment analyses, Talagrand concentration, and sophisticated OGP-based hardness arguments provides a comprehensive picture of when efficient algorithms can succeed and when inherent computational barriers arise for random GAP instances. Key contributions include (i) precise informational thresholds in sparse and dense regimes, (ii) a constructive greedy algorithm achieving near-optimal overlaps in both regimes, (iii) a rigorous 2-OGP-based barrier for stable algorithms, and (iv) a branching-OGP-based hardness result for online algorithms establishing the threshold as a fundamental computational limit in the dense regime.

Abstract

We study the graph alignment problem over two independent Erdős-Rényi graphs on vertices, with edge density falling into two regimes separated by the critical window around . 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 lies in the dense regime, resulting in a multiplicative constant factor gap between achievable and optimal solutions.
Paper Structure (24 sections, 29 theorems, 207 equations, 1 algorithm)

This paper contains 24 sections, 29 theorems, 207 equations, 1 algorithm.

Table of Contents

  1. Introduction and main results
  2. Backgrounds and related works
  3. Proof overview
  4. The informational thresholds
  5. Informational negative results
  6. Informational positive results
  7. Algorithmic positive results via greedy algorithms
  8. Framework of the algorithm
  9. Analysis of greedy algorithm: the sparse regime
  10. Analysis of greedy algorithm: the dense regime
  11. Controlling the last steps
  12. Controlling the middle steps
  13. Hardness results via the overlap-gap property
  14. Failure of stable algorithms via 2-OGP
  15. Failure of online algorithms via branching-OGP
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $(G,\mathsf G)\sim \mathbf G(n,p)^{\otimes 2}$, then the following hold. (i) When $p$ is in the sparse regime, for $S_{n,p}{=}\frac{n\log n}{\log ({\log n}/{np^2})}$, we have (ii) When $p$ is in the dense regime, for $D_{n,p}=\sqrt{n^3p^2\log n}$, we have

Theorems & Definitions (54)

  • Theorem 1.1
  • Remark 1.1
  • Theorem 1.2
  • Remark 1.2
  • Theorem 1.3: informal
  • Theorem 1.4
  • Remark 1.3
  • Proposition 2.1: Chernoff bound
  • proof : Proof of the upper bound for Theorem \ref{['thm-info']}
  • Lemma 2.2
  • ...and 44 more