Approximating $q \rightarrow p$ Norms of Non-Negative Matrices in Nearly-Linear Time
Étienne Objois, Adrian Vladu
TL;DR
The paper tackles computing the induced $\\ell_{q \rightarrow p}$-norm of non-negative matrices with $q \ge p \ge 1$, a problem with varied hardness depending on parameters. It introduces the first nearly-linear time algorithm to approximate the norm to a factor $(1-\\varepsilon)$, running in $\\tilde{O}(nnz(A)/(q\\varepsilon))$ via a coordinate-scaling, concave reformulation over the simplex and a potential-based progress framework. It also develops a width-independent and parallelizable variant, plus Lewis-weight preconditioning that yields smaller surrogate matrices and enables fast oblivious-routing constructions; a cutting-plane based scheme is provided to compute optimal oblivious routings for fixed monotone norms with runtime $\\tilde{O}(n^6 m^3)$. By connecting matrix norm approximation to oblivious routing, the work delivers practical algorithms for network optimization and robust routing, with broad implications for graph algorithms and convex-concave optimization.”
Abstract
We provide the first nearly-linear time algorithm for approximating $\ell_{q \rightarrow p}$-norms of non-negative matrices, for $q \geq p \geq 1$. Our algorithm returns a $(1-\varepsilon)$-approximation to the matrix norm in time $\widetilde{O}\left(\frac{1}{q \varepsilon} \cdot \text{nnz}(\boldsymbol{\mathit{A}})\right)$, where $\boldsymbol{\mathit{A}}$ 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-Räcke, 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 $\textit{optimal}$ 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-Räcke, and generalizes the $\ell_\infty$ routing algorithm of [Azar-Cohen-Fiat-Kaplan-Räcke, STOC '03].