Solving Linear Programs in the Current Matrix Multiplication Time

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 cu...

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Published inJournal of the ACM Vol. 68; no. 1; pp. 1 - 39
Main Authors Cohen, Michael B., Lee, Yin Tat, Song, Zhao
Format Journal Article
LanguageEnglish
Published New York Association for Computing Machinery 01.02.2021
Subjects
Algorithms
Linear programming
Mathematical analysis
Matrix methods
Multiplication
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Abstract 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 A √ W in sub-quadratic time under \ell 2 multiplicative changes in the diagonal matrix W .
AbstractList 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.
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 A √ W in sub-quadratic time under \ell 2 multiplicative changes in the diagonal matrix W .
Author Song, Zhao
Cohen, Michael B.
Lee, Yin Tat
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Snippet 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)),...
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...
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SubjectTerms Algorithms
Linear programming
Mathematical analysis
Matrix methods
Multiplication
Title Solving Linear Programs in the Current Matrix Multiplication Time
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