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.

• The paper establishes NP-completeness for optimal graph rewiring, linking spectral gap and conductance thresholds to well-known NP-hard problems.
• It demonstrates that exact optimization is intractable, justifying the reliance on heuristic approaches like SDRF and DIGL in deep GNN applications.
• The study highlights inherent trade-offs in mitigating oversmoothing versus oversquashing, underscoring the need for adaptive, architecture-aware rewiring strategies.

Complexity of Optimal Graph Rewiring for Oversmoothing and Oversquashing in GNNs

Introduction

The paper "On the Complexity of Optimal Graph Rewiring for Oversmoothing and Oversquashing in Graph Neural Networks" (2603.26140) provides a rigorous computational complexity analysis of structural graph optimization problems fundamentally linked to the degradation phenomena observed in deep GNNs. Specifically, it formulates the mitigation of oversmoothing and oversquashing as constrained graph optimization problems, with natural objective functions tied to the spectral gap and conductance, respectively. Strong NP-completeness results are established via reductions from Minimum Bisection, providing theoretical justification for the empirical reliance on heuristics rather than exact optimization.

Problem Formulation and Theoretical Framework

Oversmoothing and oversquashing in GNNs are directly determined by properties of the underlying graph:

• Oversmoothing is formalized in terms of the spectral gap (the second eigenvalue $\lambda_2$ of the normalized Laplacian), as repeated message passing through deep layers causes features to converge to the dominant eigenvector, erasing node-specific information. The problem is formulated as minimizing the second eigenvalue of the propagation matrix $P$, i.e., maximizing the spectral gap.
• Oversquashing is tied to graph bottlenecks, quantifiable by conductance ($\phi$), a classic measure for the worst-case edge expansion of vertex subsets. Low conductance implies crucial information from disparate parts of the graph is exponentially compressed through narrow cuts, resulting in information loss.

The paper introduces two decision problems:

• Graph Rewiring for Oversquashing via Conductance (GROC): Given a graph, a budget $K$ of allowed edge modifications, and a conductance threshold, decide if the graph can be rewired to achieve conductance above the threshold.
• Graph Rewiring for Oversmoothing (GROS): Given a graph, a budget $K$, and a spectral gap threshold, decide if the graph can be rewired such that the second eigenvalue of the propagation matrix is below (or spectral gap above) the given threshold.

NP-Completeness Results

The central contribution is the rigorous establishment of the NP-completeness of both GROC and GROS:

• For both problems, the paper constructs polynomial-time reductions from the Minimum Bisection problem, utilizing expander graph embeddings and careful instance scaling to preserve the hardness of the original problem in the new objective domain.
• The embedding ensures that the conductance (or spectral gap) in the constructed graph directly reflects the solution quality of the bisection problem in the original input.
• Both decision versions are shown to reside in NP by appealing to the tractability of verifying conductance (via min-cut/max-flow algorithms) and eigenvalue inequalities (via characteristic polynomials and root isolation), given a candidate solution.

This demonstrates the computational intractability of finding optimal rewirings for either oversmoothing or oversquashing mitigation, even if only a finite number of edges can be changed, and even when the property to be optimized is as mathematically tractable as conductance or spectral gap.

Implications for Practical Graph Rewiring

The intractability results clarify why all practical GNN graph rewiring and augmentation methods—such as SDRF (discrete Ricci flow) and DIGL (diffusion-based strategies)—must fundamentally rely on heuristics or (at best) approximation algorithms:

• The complexity analysis of SDRF and DIGL shows they admit polynomial-time implementations (e.g., $O(n^2)$ per iteration), but no polynomial-time exact optimality guarantees for oversmoothing/oversquashing are possible under standard complexity-theoretic assumptions.
• Theoretical hardness of approximation is also discussed: known results imply strong inapproximability for both conductance and the second eigenvalue in regular graphs, precluding efficient algorithms with non-trivial worst-case guarantees.

This formalizes the empirical observation that current successful rewiring methods trade off between increased local expansion (to mitigate oversquashing) and the risk of oversmoothing induced by denser spectra.

Theoretical and Practical Implications

The analysis reveals a structural duality between oversmoothing and oversquashing: optimizing for one results in tradeoffs for the other due to the Cheeger inequality linking spectral gap and conductance in regular graphs. This yields an inherent tension where optimal mitigation of both cannot be achieved simultaneously except in dense random-like graphs, which generally remain inappropriate for most real-world settings.

From a practical perspective, the findings:

• Justify heuristic strategies that incrementally add expansion while controlling for spectral concentration.
• Highlight the importance of architecture-level mitigations (residual connections, normalization schemes) in conjunction with structural rewiring for scalable deep GNNs.
• Underscore the likely necessity for adaptive task-aware rewiring approaches rather than brute-force spectral or conductance maximization.
• Indicate the potential value of hybrid frameworks combining algorithmic graph augmentations with co-learned or end-to-end optimized architectures.

On the theoretical side, these results chart clear boundaries for the class of feasible graph structure optimizations: any approach promising exact or even constant-factor optimal rewiring for these GNN pathologies is infeasible unless P=NP.

Future Directions

Key open directions suggested by the results include:

• Extending the complexity analysis to weighted, directed, or signed graphs, reflecting nuances of real-world graph structures.
• Considering rewirings that are explicitly task-dependent or which incorporate labels and side-information.
• Analyzing architectural (not just structural) mitigation for oversmoothing and oversquashing, particularly for attention-based or higher-order GNN variants.
• Investigating empirical instances to determine the typical versus worst-case hardness and efficacy of heuristic rewiring in practice.

Conclusion

This work establishes the fundamental computational intractability of optimizing graph topology to mitigate oversmoothing and oversquashing in GNNs, rigorously proving the NP-completeness of such rewiring even for standard objective functions (conductance and spectral gap). These results provide a strong mathematical basis explaining the dominance of heuristic methods in practical GNN applications and delineate the theoretical limits for future advances in graph-topological optimization for GNNs (2603.26140).