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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Veröffentlicht in: | Journal of the ACM 2021-02, Vol.68 (1), p.1-39 |
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Hauptverfasser: | , , |
Format: | Artikel |
Sprache: | eng |
Schlagworte: | |
Online-Zugang: | Volltext |
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Zusammenfassung: | 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
. |
---|---|
ISSN: | 0004-5411 1557-735X |
DOI: | 10.1145/3424305 |