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On the Complexity of Optimal Graph Rewiring for Oversmoothing and Oversquashing in Graph Neural Networks

Mostafa Haghir Chehreghani

Abstract

Graph Neural Networks (GNNs) face two fundamental challenges when scaled to deep architectures: oversmoothing, where node representations converge to indistinguishable vectors, and oversquashing, where information from distant nodes fails to propagate through bottlenecks. Both phenomena are intimately tied to the underlying graph structure, raising a natural question: can we optimize the graph topology to mitigate these issues? This paper provides a theoretical investigation of the computational complexity of such graph structure optimization. We formulate oversmoothing and oversquashing mitigation as graph optimization problems based on spectral gap and conductance, respectively. We prove that exact optimization for either problem is NP-hard through reductions from Minimum Bisection, establishing NP-completeness of the decision versions. Our results provide theoretical foundations for understanding the fundamental limits of graph rewiring for GNN optimization and justify the use of approximation algorithms and heuristic methods in practice.

On the Complexity of Optimal Graph Rewiring for Oversmoothing and Oversquashing in Graph Neural Networks

Abstract

Graph Neural Networks (GNNs) face two fundamental challenges when scaled to deep architectures: oversmoothing, where node representations converge to indistinguishable vectors, and oversquashing, where information from distant nodes fails to propagate through bottlenecks. Both phenomena are intimately tied to the underlying graph structure, raising a natural question: can we optimize the graph topology to mitigate these issues? This paper provides a theoretical investigation of the computational complexity of such graph structure optimization. We formulate oversmoothing and oversquashing mitigation as graph optimization problems based on spectral gap and conductance, respectively. We prove that exact optimization for either problem is NP-hard through reductions from Minimum Bisection, establishing NP-completeness of the decision versions. Our results provide theoretical foundations for understanding the fundamental limits of graph rewiring for GNN optimization and justify the use of approximation algorithms and heuristic methods in practice.

Paper Structure

This paper contains 20 sections, 4 theorems, 17 equations.

Table of Contents

  1. Keywords:
  2. Introduction
  3. Contributions
  4. Related Work
  5. Preliminaries
  6. Basic Graph Notation
  7. Spectral Graph Theory and Conductance
  8. Graph Rewiring for Oversquashing via Conductance
  9. Problem Formulation
  10. Expander Embedding
  11. Pre‑processing Minimum Bisection Instances
  12. NP-Completeness of Conductance Maximization
  13. Graph Rewiring for Oversmoothing
  14. Spectral Formulation
  15. NP-Completeness of Oversmoothing Minimization
  16. ...and 5 more sections

Key Result

Lemma 2

There exists a polynomial‑time algorithm that, given a graph $H = (V, E_H)$ with $n$ vertices (assumed large enough, $n \ge n_0$), outputs a $3$-regular graph $G = (V', E_G)$ with $|V'| = 2n$ and constants $c_1, c_2 > 0$, $c_3 \in (0,1)$ satisfying: The constants $c_1, c_2, c_3$ can be made arbitrarily small (or large) by choosing a sufficiently strong (or weak) expander.

Theorems & Definitions (8)

  • Definition 1: Graph Rewiring for Oversquashing via Conductance (GROC)
  • Lemma 2: Expander Embedding
  • Lemma 3: Instance Scaling
  • Theorem 4
  • proof
  • Definition 5: Graph Rewiring for Oversmoothing (GROS)
  • Theorem 6
  • proof