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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| container_title | Journal of the ACM |
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| creator | Cohen, Michael B. Lee, Yin Tat Song, Zhao |
| description | 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
. |
| doi_str_mv | 10.1145/3424305 |
| format | Article |
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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
.</description><identifier>ISSN: 0004-5411</identifier><identifier>EISSN: 1557-735X</identifier><identifier>DOI: 10.1145/3424305</identifier><language>eng</language><publisher>New York: Association for Computing Machinery</publisher><subject>Algorithms ; Linear programming ; Mathematical analysis ; Matrix methods ; Multiplication</subject><ispartof>Journal of the ACM, 2021-02, Vol.68 (1), p.1-39</ispartof><rights>Copyright Association for Computing Machinery Feb 2021</rights><lds50>peer_reviewed</lds50><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c286t-ccfbb3b062757bb7276ba7d4e6e2589505061a60d1428b4d2faa6c659431e4bd3</citedby><cites>FETCH-LOGICAL-c286t-ccfbb3b062757bb7276ba7d4e6e2589505061a60d1428b4d2faa6c659431e4bd3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><link.rule.ids>314,780,784,27922,27923</link.rule.ids></links><search><creatorcontrib>Cohen, Michael B.</creatorcontrib><creatorcontrib>Lee, Yin Tat</creatorcontrib><creatorcontrib>Song, Zhao</creatorcontrib><title>Solving Linear Programs in the Current Matrix Multiplication Time</title><title>Journal of the ACM</title><description>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
.</description><subject>Algorithms</subject><subject>Linear programming</subject><subject>Mathematical analysis</subject><subject>Matrix methods</subject><subject>Multiplication</subject><issn>0004-5411</issn><issn>1557-735X</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2021</creationdate><recordtype>article</recordtype><recordid>eNotkF1LwzAYhYMoWKf4FwJeeFV9891ejqJT2FBwgncladOZ0S-TVPTfW9muDgcezoEHoWsCd4Rwcc845QzECUqIECpVTHycogQAeCo4IefoIoT9XIGCStDybWi_Xb_Da9db7fGrH3ZedwG7HsdPi4vJe9tHvNHRux-8mdroxtZVOrqhx1vX2Ut01ug22KtjLtD748O2eErXL6vnYrlOK5rJmFZVYwwzIKkSyhhFlTRa1dxKS0WWCxAgiZZQE04zw2vaaC0rKXLOiOWmZgt0c9gd_fA12RDL_TD5fr4sKc9zoDLL1UzdHqjKDyF425Sjd532vyWB8t9PefTD_gBdqFX7</recordid><startdate>20210201</startdate><enddate>20210201</enddate><creator>Cohen, Michael B.</creator><creator>Lee, Yin Tat</creator><creator>Song, Zhao</creator><general>Association for Computing Machinery</general><scope>AAYXX</scope><scope>CITATION</scope><scope>7SC</scope><scope>8FD</scope><scope>JQ2</scope><scope>L7M</scope><scope>L~C</scope><scope>L~D</scope></search><sort><creationdate>20210201</creationdate><title>Solving Linear Programs in the Current Matrix Multiplication Time</title><author>Cohen, Michael B. ; Lee, Yin Tat ; Song, Zhao</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c286t-ccfbb3b062757bb7276ba7d4e6e2589505061a60d1428b4d2faa6c659431e4bd3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2021</creationdate><topic>Algorithms</topic><topic>Linear programming</topic><topic>Mathematical analysis</topic><topic>Matrix methods</topic><topic>Multiplication</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Cohen, Michael B.</creatorcontrib><creatorcontrib>Lee, Yin Tat</creatorcontrib><creatorcontrib>Song, Zhao</creatorcontrib><collection>CrossRef</collection><collection>Computer and Information Systems Abstracts</collection><collection>Technology Research Database</collection><collection>ProQuest Computer Science Collection</collection><collection>Advanced Technologies Database with Aerospace</collection><collection>Computer and Information Systems Abstracts Academic</collection><collection>Computer and Information Systems Abstracts Professional</collection><jtitle>Journal of the ACM</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Cohen, Michael B.</au><au>Lee, Yin Tat</au><au>Song, Zhao</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Solving Linear Programs in the Current Matrix Multiplication Time</atitle><jtitle>Journal of the ACM</jtitle><date>2021-02-01</date><risdate>2021</risdate><volume>68</volume><issue>1</issue><spage>1</spage><epage>39</epage><pages>1-39</pages><issn>0004-5411</issn><eissn>1557-735X</eissn><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
.</abstract><cop>New York</cop><pub>Association for Computing Machinery</pub><doi>10.1145/3424305</doi><tpages>39</tpages><oa>free_for_read</oa></addata></record> |
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| ispartof | Journal of the ACM, 2021-02, Vol.68 (1), p.1-39 |
| issn | 0004-5411 1557-735X |
| language | eng |
| recordid | cdi_proquest_journals_2499026897 |
| source | ACM Digital Library Complete |
| subjects | Algorithms Linear programming Mathematical analysis Matrix methods Multiplication |
| title | Solving Linear Programs in the Current Matrix Multiplication Time |
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