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

On the Complexity of Optimal Graph Rewiring for Oversmoothing and Oversquashing in Graph Neural Networks | Signal Canvas | ScienceToStartup

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Canonical route: /signal-canvas/on-the-complexity-of-optimal-graph-rewiring-for-oversmoothing-and-oversquashing-in-graph-neural-networks

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Proof freshness: stale
Proof status: unverified
Display score: 2/10
Last proof check: 2026-03-30
Score updated: 2026-04-02
Score fresh until: 2026-05-02
References: 25
Source count: 3
Coverage: 50%

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Evidence Receipt

Route status: building

Claims: 7

References: 25

Proof: Verification pending

Freshness state: computing

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

PDF: https://arxiv.org/pdf/2603.26140v1

Source count: 3

Coverage: 50%

Last proof check: 2026-03-30T22:00:24.640Z

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

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Ignore

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Claim map

Strong 7Mixed 0Weak 0

Evidencepartial
We prove that exact optimization for either problem is NP-hard through reductions from Minimum Bisection, establishing NP-completeness of the decision versions.
Implicationpartial
The abstract explicitly states this and the analysis mentions reductions from Minimum Bisection to prove NP-hardness for oversmoothing mitigation.
Verificationpartial
partial
Evidencepartial
We prove that exact optimization for either problem is NP-hard through reductions from Minimum Bisection, establishing NP-completeness of the decision versions.
Implicationpartial
The abstract explicitly states this and the analysis mentions reductions from Minimum Bisection to prove NP-hardness for oversquashing mitigation.
Verificationpartial
partial
Evidencepartial
GROC is NP-hard, and with membership in NP it is NP-complete.
Implicationpartial
The paper explicitly defines GROS and states its NP-hardness, supported by reductions from Minimum Bisection.
Verificationpartial
partial
Evidencepartial
However, the fundamental computational complexity of such graph optimization problems has remained unexplored.
Implicationpartial
The abstract and introduction highlight this as a gap that the paper addresses, stating 'the fundamental computational complexity of such graph optimization problems has remained unexplored.'
Verificationpartial
partial
Evidencepartial
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.
Implicationpartial
The abstract and analysis explicitly state that the NP-hardness results justify the use of these methods.
Verificationpartial
partial
Evidencepartial
For a regular graph, the symmetric normalized propagation matrixP = D−1/2AD−1/2 has eigenvalues 1 = µ1(P ) ≥µ 2(P ) ≥ · · · ≥µ n(P ) ≥ −1. The Dirichlet energy after L layers decays as E(X (L)) ≤ (µ2(P ))2LE(X (0)). Thus, to prevent over-smoothing, we wish to minimize µ2(P ).
Implicationpartial
The paper formulates oversmoothing mitigation based on spectral gap and explains the relationship between the spectral gap and Dirichlet energy decay.
Verificationpartial
partial
Evidencepartial
We formulate oversmoothing and oversquashing mitigation as graph optimization problems based on spectral gap and conductance, respectively.
Implicationpartial
The paper formulates oversquashing mitigation as a problem based on conductance.
Verificationpartial
partial

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

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