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Approximating q → p Norms of Non-Negative Matrices in Nearly-Linear Time

Authors Etienne Objois , Adrian Vladu

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LIPIcs.STACS.2026.69.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Routing and network design problems
Keywords
  • matrix norm
  • Perron-Frobenius theory
  • oblivious routings
  • input-sparsity time
  • lp norm

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Abstract

We provide the first nearly-linear time algorithm for approximating 𝓁_{q → p}-norms of non-negative matrices, for q ≥ p ≥ 1. Our algorithm returns a (1-ε)-approximation to the matrix norm in time Õ(1/(q ε) ⋅ nnz(A)), where A is the input matrix, and improves upon the previous state of the art, which either proved convergence only in the limit [Boyd '74], or had very high polynomial running times [Bhaskara-Vijayraghavan, SODA '11]. Our algorithm is extremely simple, and is largely inspired from the coordinate-scaling approach used for positive linear program solvers. Our algorithm can readily be used in the [Englert-Räcke, FOCS '09] to improve the running time of constructing O(log n)-competitive 𝓁_p-oblivious routings.

Cite As Get BibTex

Etienne Objois and Adrian Vladu. Approximating q → p Norms of Non-Negative Matrices in Nearly-Linear Time. In 43rd International Symposium on Theoretical Aspects of Computer Science (STACS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 364, pp. 69:1-69:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.STACS.2026.69

Author Details

Etienne Objois
  • IRIF, Université Paris Cité, France
Adrian Vladu
  • CNRS, IRIF, Université Paris Cité, France

Funding

This work was partially supported by the French Agence Nationale de la Recherche (ANR), under grant ANR-21-CE48-0016 (project COMCOPT).

Acknowledgements

We thank Alina Ene and Huy Lê Nguy~ên for helpful conversations on approximating matrix norms.

References

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