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Switching Graph Matrix Norm Bounds: From i.i.d. to Random Regular Graphs

Author Jeff Xu

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  • Theory of computation → Semidefinite programming
Keywords
  • Semidefinite programming
  • random matrices
  • average-case complexity

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Abstract

In this work, we give novel spectral norm bounds for graph matrix on inputs being random regular graphs. Graph matrix is a family of random matrices with entries given by polynomial functions of the underlying input. These matrices have been known to be the backbone for the analysis of various average-case algorithms and hardness. Previous investigations of such matrices are largely restricted to the Erdős-Rényi model, and tight matrix norm bounds on regular graphs are only known for specific examples. We unite these two lines of investigations, and give the first result departing from the Erdős-Rényi setting in the full generality of graph matrices. We believe our norm bound result would enable a simple transfer of spectral analysis for average-case algorithms and hardness between these two distributions of random graphs.
As an application of our spectral norm bounds, we show that higher-degree Sum-of-Squares lower bounds for the independent set problem on Erdős-Rényi random graphs can be switched into lower bounds on random d-regular graphs. Our main conceptual insight is that existing Sum-of-Squares lower bounds analysis based on moment methods are surprisingly robust, and amenable for a light-weight translation. Our result is the first to address the general open question of analyzing higher-degree Sum-of-Squares on random regular graphs.

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Jeff Xu. Switching Graph Matrix Norm Bounds: From i.i.d. to Random Regular Graphs. In 40th Computational Complexity Conference (CCC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 339, pp. 11:1-11:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.CCC.2025.11

Author Details

Jeff Xu
  • Carnegie Mellon University, Pittsburgh, PA, USA

Funding

  • Xu, Jeff: Supported in part by NSF CAREER Award #2047933.

Acknowledgements

We would like to thank Pravesh Kothari, Madhur Tulsiani and Aaron Potechin for various discussions. We thank the anonymous reviewer for pointing out that our particular SoS lower bound application in the regime of poly-logarithmic average-degree can also be obtained from Kim-Vu’s sandwich conjecture, and all reviewers for their helpful suggestions for improving our writing.

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