Solving Linear Programs in the Current Matrix Multiplication Time

TL;DR

This paper introduces a new algorithm for solving linear programs efficiently by leveraging matrix multiplication exponents, achieving near-optimal time complexity and introducing novel concepts like a stochastic central path method.

Contribution

The paper presents a novel algorithm for linear programming that operates in current matrix multiplication time, utilizing a stochastic central path method and efficient projection matrix maintenance.

Findings

Achieves $O^*(n^{ ext{omega}} ext{log}(n/ extdelta))$ time with current matrix multiplication exponents.

Introduces a stochastic central path method for linear programming.

Develops a sub-quadratic time method for maintaining specific projection matrices.

Abstract

This paper shows how to solve linear programs of the form $\min_{Ax=b,x\geq0} c^\top x$ with $n$ variables in time $$O^*((n^{\omega}+n^{2.5-\alpha/2}+n^{2+1/6}) \log(n/\delta))$$ where $\omega$ is the exponent of matrix multiplication, $\alpha$ is the dual exponent of matrix multiplication, and $\delta$ is the relative accuracy. For the current value of $\omega\sim2.37$ and $\alpha\sim0.31$, our algorithm takes $O^*(n^{\omega} \log(n/\delta))$ time. When $\omega = 2$, our algorithm takes $O^*(n^{2+1/6} \log(n/\delta))$ time. Our algorithm utilizes several new concepts that we believe may be of independent interest: $\bullet$ We define a stochastic central path method. $\bullet$ We show how to maintain a projection matrix $\sqrt{W}A^{\top}(AWA^{\top})^{-1}A\sqrt{W}$ in sub-quadratic time under $\ell_{2}$ multiplicative changes in the diagonal matrix $W$.