Total dual dyadicness and dyadic generating sets
A vector is dyadic if each of its entries is a dyadic rational number, i.e. of the form a 2 k for some integers a , k with k ≥ 0 . A linear system A x ≤ b with integral data is totally dual dyadic if whenever min { b ⊤ y : A ⊤ y = w , y ≥ 0 } for w integral, has an optimal solution, it has a dyadic...
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| Veröffentlicht in: | Mathematical programming 2024-07, Vol.206 (1-2), p.125-143 |
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| container_title | Mathematical programming |
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| creator | Abdi, Ahmad Cornuéjols, Gérard Guenin, Bertrand Tunçel, Levent |
| description | A vector is
dyadic
if each of its entries is a dyadic rational number, i.e. of the form
a
2
k
for some integers
a
,
k
with
k
≥
0
. A linear system
A
x
≤
b
with integral data is
totally dual dyadic
if whenever
min
{
b
⊤
y
:
A
⊤
y
=
w
,
y
≥
0
}
for
w
integral, has an optimal solution, it has a dyadic optimal solution. In this paper, we study total dual dyadicness, and give a co-NP characterization of it in terms of
dyadic generating sets for cones and subspaces
, the former being the dyadic analogue of
Hilbert bases
, and the latter a polynomial-time recognizable relaxation of the former. Along the way, we see some surprising turn of events when compared to total dual integrality, primarily led by the
density
of the dyadic rationals. Our study ultimately leads to a better understanding of total dual integrality and polyhedral integrality. We see examples from dyadic matroids,
T
-joins, cycles, and perfect matchings of a graph. |
| doi_str_mv | 10.1007/s10107-023-01967-z |
| format | Article |
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dyadic
if each of its entries is a dyadic rational number, i.e. of the form
a
2
k
for some integers
a
,
k
with
k
≥
0
. A linear system
A
x
≤
b
with integral data is
totally dual dyadic
if whenever
min
{
b
⊤
y
:
A
⊤
y
=
w
,
y
≥
0
}
for
w
integral, has an optimal solution, it has a dyadic optimal solution. In this paper, we study total dual dyadicness, and give a co-NP characterization of it in terms of
dyadic generating sets for cones and subspaces
, the former being the dyadic analogue of
Hilbert bases
, and the latter a polynomial-time recognizable relaxation of the former. Along the way, we see some surprising turn of events when compared to total dual integrality, primarily led by the
density
of the dyadic rationals. Our study ultimately leads to a better understanding of total dual integrality and polyhedral integrality. We see examples from dyadic matroids,
T
-joins, cycles, and perfect matchings of a graph.</description><identifier>ISSN: 0025-5610</identifier><identifier>EISSN: 1436-4646</identifier><identifier>DOI: 10.1007/s10107-023-01967-z</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Calculus of Variations and Optimal Control; Optimization ; Combinatorics ; Full Length Paper ; Linear systems ; Mathematical and Computational Physics ; Mathematical Methods in Physics ; Mathematics ; Mathematics and Statistics ; Mathematics of Computing ; Numerical Analysis ; Polynomials ; Subspaces ; Theoretical</subject><ispartof>Mathematical programming, 2024-07, Vol.206 (1-2), p.125-143</ispartof><rights>Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2023. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-3cf0e27931c72759f1a247eecc667f00d801f3f62fd93c51aec6a6e858bd26583</cites><orcidid>0000-0002-3976-1021</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s10107-023-01967-z$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s10107-023-01967-z$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Abdi, Ahmad</creatorcontrib><creatorcontrib>Cornuéjols, Gérard</creatorcontrib><creatorcontrib>Guenin, Bertrand</creatorcontrib><creatorcontrib>Tunçel, Levent</creatorcontrib><title>Total dual dyadicness and dyadic generating sets</title><title>Mathematical programming</title><addtitle>Math. Program</addtitle><description>A vector is
dyadic
if each of its entries is a dyadic rational number, i.e. of the form
a
2
k
for some integers
a
,
k
with
k
≥
0
. A linear system
A
x
≤
b
with integral data is
totally dual dyadic
if whenever
min
{
b
⊤
y
:
A
⊤
y
=
w
,
y
≥
0
}
for
w
integral, has an optimal solution, it has a dyadic optimal solution. In this paper, we study total dual dyadicness, and give a co-NP characterization of it in terms of
dyadic generating sets for cones and subspaces
, the former being the dyadic analogue of
Hilbert bases
, and the latter a polynomial-time recognizable relaxation of the former. Along the way, we see some surprising turn of events when compared to total dual integrality, primarily led by the
density
of the dyadic rationals. Our study ultimately leads to a better understanding of total dual integrality and polyhedral integrality. We see examples from dyadic matroids,
T
-joins, cycles, and perfect matchings of a graph.</description><subject>Calculus of Variations and Optimal Control; Optimization</subject><subject>Combinatorics</subject><subject>Full Length Paper</subject><subject>Linear systems</subject><subject>Mathematical and Computational Physics</subject><subject>Mathematical Methods in Physics</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Mathematics of Computing</subject><subject>Numerical Analysis</subject><subject>Polynomials</subject><subject>Subspaces</subject><subject>Theoretical</subject><issn>0025-5610</issn><issn>1436-4646</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kE9LAzEQxYMoWKtfwNOC5-hMZjfJHqX4Dwpe6jnE7KS01N2abA_tp3frFrx5mWHgvTe8nxC3CPcIYB4yAoKRoEgC1trIw5mYYElalrrU52ICoCpZaYRLcZXzGgCQrJ0IWHS93xTN7jj2vlmFlnMufNuczmLJLSffr9plkbnP1-Ii-k3mm9Oeio_np8XsVc7fX95mj3MZlIFeUojAytSEwShT1RG9Kg1zCFqbCNBYwEhRq9jUFCr0HLTXbCv72ShdWZqKuzF3m7rvHeferbtdaoeXjsCWZIiGClOhRlVIXc6Jo9um1ZdPe4fgjmTcSMYNZNwvGXcYTDSa8iBul5z-ov9x_QAfNGWi</recordid><startdate>20240701</startdate><enddate>20240701</enddate><creator>Abdi, Ahmad</creator><creator>Cornuéjols, Gérard</creator><creator>Guenin, Bertrand</creator><creator>Tunçel, Levent</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</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><orcidid>https://orcid.org/0000-0002-3976-1021</orcidid></search><sort><creationdate>20240701</creationdate><title>Total dual dyadicness and dyadic generating sets</title><author>Abdi, Ahmad ; Cornuéjols, Gérard ; Guenin, Bertrand ; Tunçel, Levent</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-3cf0e27931c72759f1a247eecc667f00d801f3f62fd93c51aec6a6e858bd26583</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Calculus of Variations and Optimal Control; Optimization</topic><topic>Combinatorics</topic><topic>Full Length Paper</topic><topic>Linear systems</topic><topic>Mathematical and Computational Physics</topic><topic>Mathematical Methods in Physics</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Mathematics of Computing</topic><topic>Numerical Analysis</topic><topic>Polynomials</topic><topic>Subspaces</topic><topic>Theoretical</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Abdi, Ahmad</creatorcontrib><creatorcontrib>Cornuéjols, Gérard</creatorcontrib><creatorcontrib>Guenin, Bertrand</creatorcontrib><creatorcontrib>Tunçel, Levent</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>Mathematical programming</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Abdi, Ahmad</au><au>Cornuéjols, Gérard</au><au>Guenin, Bertrand</au><au>Tunçel, Levent</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Total dual dyadicness and dyadic generating sets</atitle><jtitle>Mathematical programming</jtitle><stitle>Math. Program</stitle><date>2024-07-01</date><risdate>2024</risdate><volume>206</volume><issue>1-2</issue><spage>125</spage><epage>143</epage><pages>125-143</pages><issn>0025-5610</issn><eissn>1436-4646</eissn><abstract>A vector is
dyadic
if each of its entries is a dyadic rational number, i.e. of the form
a
2
k
for some integers
a
,
k
with
k
≥
0
. A linear system
A
x
≤
b
with integral data is
totally dual dyadic
if whenever
min
{
b
⊤
y
:
A
⊤
y
=
w
,
y
≥
0
}
for
w
integral, has an optimal solution, it has a dyadic optimal solution. In this paper, we study total dual dyadicness, and give a co-NP characterization of it in terms of
dyadic generating sets for cones and subspaces
, the former being the dyadic analogue of
Hilbert bases
, and the latter a polynomial-time recognizable relaxation of the former. Along the way, we see some surprising turn of events when compared to total dual integrality, primarily led by the
density
of the dyadic rationals. Our study ultimately leads to a better understanding of total dual integrality and polyhedral integrality. We see examples from dyadic matroids,
T
-joins, cycles, and perfect matchings of a graph.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s10107-023-01967-z</doi><tpages>19</tpages><orcidid>https://orcid.org/0000-0002-3976-1021</orcidid></addata></record> |
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| ispartof | Mathematical programming, 2024-07, Vol.206 (1-2), p.125-143 |
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| language | eng |
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| subjects | Calculus of Variations and Optimal Control Optimization Combinatorics Full Length Paper Linear systems Mathematical and Computational Physics Mathematical Methods in Physics Mathematics Mathematics and Statistics Mathematics of Computing Numerical Analysis Polynomials Subspaces Theoretical |
| title | Total dual dyadicness and dyadic generating sets |
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