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Solving Linear Programs in the Current Matrix Multiplication Time
Journal of the ACM ( IF 2.3 ) Pub Date : 2021-01-05 , DOI: 10.1145/3424305
Michael B. Cohen 1 , Yin Tat Lee 2 , Zhao Song 3
Affiliation  

This article shows how to solve linear programs of the form min Ax = b , x ≥ 0 c x with n variables in time O * (( n ω + n 2.5−α/2 + n 2+1/6 ) log ( n /δ)), where ω is the exponent of matrix multiplication, α is the dual exponent of matrix multiplication, and δ is the relative accuracy. For the current value of ω δ 2.37 and α δ 0.31, our algorithm takes O * ( n ω log ( n /δ)) time. When ω = 2, our algorithm takes O * ( n 2+1/6 log ( n /δ)) time. Our algorithm utilizes several new concepts that we believe may be of independent interest: • We define a stochastic central path method. • We show how to maintain a projection matrix √ W A ( AWA ) −1 AW in sub-quadratic time under \ell 2 multiplicative changes in the diagonal matrix W .

中文翻译:

在当前矩阵乘法时间内求解线性规划

本文展示了如何求解形式为 min 的线性规划 斧头=b,X≥ 0 C Xn时间变量 *((n ω+n 2.5−α/2+n 2+1/6) 日志 (n/δ)),其中ω是矩阵乘法的指数,α是矩阵乘法的对偶指数,δ是相对精度。对于 ω δ 2.37 和 α δ 0.31 的当前值,我们的算法取 *(n ω日志 (n/δ)) 时间。当 ω = 2 时,我们的算法取 *(n 2+1/6日志 (n/δ)) 时间。我们的算法利用了几个我们认为可能具有独立意义的新概念: • 我们定义了一种随机中心路径方法。• 我们展示了如何维护一个投影矩阵 √W 一种 (阿瓦 )-1 一种W在 \ell 下的次二次时间2对角矩阵的乘法变化W.
更新日期:2021-01-05
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