Solving Linear Programs in the Current Matrix Multiplication Time
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3424305.pdf
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719.78 KB
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Adobe PDF
Checksum (MD5)
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Author(s)
Cohen, Michael
Lee, Yin Tat
Song, Zhao
Date Issued
January 5, 2021
Journal
Journal of the ACM
Publisher
Association for Computing Machinery
Citation
Michael B. Cohen, Yin Tat Lee, and Zhao Song. 2021. Solving Linear Programs in the Current Matrix Multiplication Time. J. ACM 68, 1, Article 3 (February 2021), 39 pages.
Version
Final published version
Abstract
This article shows how to solve linear programs of the form minAx=b,x≥ 0 c⊤ x with n variables in time O*((nω+n2.5−α/2+n2+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*(nω log (n/δ)) time. When ω = 2, our algorithm takes O*(n2+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 √ WA⊤ (AWA⊤)−1A√ W in sub-quadratic time under \ell2 multiplicative changes in the diagonal matrix W.
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Article is made available in accordance with the publisher's policy and may be subject to US copyright law. Please refer to the publisher's site for terms of use.
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DOI of Published Version
https://doi.org/10.1145/3424305
