Faster Dynamic Matrix Inverse for Faster LPs

TL;DR

This paper introduces a new dynamic matrix inverse data structure that accelerates linear programming solvers by enabling faster low-rank updates, leading to the fastest known dense LP algorithms with improved theoretical running times.

Contribution

The paper presents a novel recursive data structure for dynamic matrix inversion using low-rank updates and sketching, significantly improving LP solver efficiency.

Findings

Achieves faster amortized update times for matrix inverse maintenance.

Reduces LP solver complexity from O*(n^{2.16}) to approximately O*(n^{2.055}).

Introduces techniques potentially applicable to broader dynamic matrix problems.

Abstract

Motivated by recent Linear Programming solvers, we design dynamic data structures for maintaining the inverse of an $n\times n$ real matrix under $\textit{low-rank}$ updates, with polynomially faster amortized running time. Our data structure is based on a recursive application of the Woodbury-Morrison identity for implementing $\textit{cascading}$ low-rank updates, combined with recent sketching technology. Our techniques and amortized analysis of multi-level partial updates, may be of broader interest to dynamic matrix problems. This data structure leads to the fastest known LP solver for general (dense) linear programs, improving the running time of the recent algorithms of (Cohen et al.'19, Lee et al.'19, Brand'20) from $O^*(n^{2+ \max\{\frac{1}{6}, \omega-2, \frac{1-\alpha}{2}\}})$ to $O^*(n^{2+\max\{\frac{1}{18}, \omega-2, \frac{1-\alpha}{2}\}})$, where $\omega$ and $\alpha$ are the fast matrix multiplication exponent and its dual. Hence, under the common belief that $\omega \approx 2$ and $\alpha \approx 1$, our LP solver runs in $O^*(n^{2.055})$ time instead of $O^*(n^{2.16})$.