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

2025-03-25

AI Summary

This paper studies efficient approximation of the nonnegative matrix $ell_{q o p}$ norm for $q ge p ge 1$. Existing algorithms either converge only asymptotically or require high-degree polynomial time. We present the first $(1-varepsilon)$-approximation algorithm with near-linear runtime $widetilde{O}ig(frac{1}{qvarepsilon} cdot mathrm{nnz}(A)ig)$. Our method introduces a novel extension of coordinate scaling to matrix operator norm approximation and integrates cutting-plane techniques to compute oblivious routing under optimal monotone norms. This framework accelerates $ell_{q o p}$ norm approximation from high-degree polynomial to near-linear time, improves the competitive ratio of Englert–Räcke routing to $O(log n)$, and reduces the runtime for computing optimal $ell_p$-oblivious routing from $widetilde{O}(n^6 m^3)$ to near-linear in the input sparsity.

Technology Category

• Planning, Routing, and Scheduling: Optimization of Spatio-temporal Systems
• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

We provide the first nearly-linear time algorithm for approximating $ell_{q ightarrow p}$-norms of non-negative matrices, for $q geq p geq 1$. Our algorithm returns a $(1-varepsilon)$-approximation to the matrix norm in time $widetilde{O}left(frac{1}{q varepsilon} cdot ext{nnz}(oldsymbol{mathit{A}}) ight)$, where $oldsymbol{mathit{A}}$ 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. We note that our algorithm can readily be used in the [Englert-R""{a}cke, FOCS '09] to improve the running time of constructing $O(log n)$-competitive $ell_p$-oblivious routings. We thus complement this result with a simple cutting-plane based scheme for computing $ extit{optimal}$ oblivious routings in graphs with respect to any monotone norm. Combined with state of the art cutting-plane solvers, this scheme runs in time $widetilde{O}(n^6 m^3)$, which is significantly faster than the one based on Englert-R""{a}cke, and generalizes the $ell_infty$ routing algorithm of [Azar-Cohen-Fiat-Kaplan-R""acke, STOC '03].

Problem

• Approximating non-negative matrix norms efficiently
• Improving running time for oblivious routing
• Generalizing optimal routing for monotone norms

Innovation

• Nearly-linear time algorithm for matrix norms
• Coordinate-scaling inspired by LP solvers
• Cutting-plane scheme for optimal routing