About the Structure of the Integer Cone and Its Application to Bin Packing
We consider the bin packing problem with d different item sizes and revisit the structure theorem given by Goemans and Rothvoß about solutions of the integer cone. We present new techniques on how solutions can be modified and give a new structure theorem that relies on the set of vertices of the un...
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| Veröffentlicht in: | Mathematics of operations research 2020-11, Vol.45 (4), p.1498-1511 |
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| container_title | Mathematics of operations research |
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| creator | Jansen, Klaus Klein, Kim-Manuel |
| description | We consider the bin packing problem with
d
different item sizes and revisit the structure theorem given by Goemans and Rothvoß about solutions of the integer cone. We present new techniques on how solutions can be modified and give a new structure theorem that relies on the set of vertices of the underlying integer polytope. As a result of our new structure theorem, we obtain an algorithm for the bin packing problem with running time
|
V
|
2
O
(
d
)
⋅
e
n
c
(
I
)
O
(
1
)
, where
V
is the set of vertices of the integer knapsack polytope, and
e
n
c
(
I
)
is the encoding length of the bin packing instance. The algorithm is fixed-parameter tractable, parameterized by the number of vertices of the integer knapsack polytope
|
V
|
. This shows that the bin packing problem can be solved efficiently when the underlying integer knapsack polytope has an easy structure (i.e., has a small number of vertices). Furthermore, we show that the presented bounds of the structure theorem are asymptotically tight. We give a construction of bin packing instances using new structural insights and classical number theoretical theorems which yield the desired lower bound. |
| doi_str_mv | 10.1287/moor.2019.1040 |
| format | Article |
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d
different item sizes and revisit the structure theorem given by Goemans and Rothvoß about solutions of the integer cone. We present new techniques on how solutions can be modified and give a new structure theorem that relies on the set of vertices of the underlying integer polytope. As a result of our new structure theorem, we obtain an algorithm for the bin packing problem with running time
|
V
|
2
O
(
d
)
⋅
e
n
c
(
I
)
O
(
1
)
, where
V
is the set of vertices of the integer knapsack polytope, and
e
n
c
(
I
)
is the encoding length of the bin packing instance. The algorithm is fixed-parameter tractable, parameterized by the number of vertices of the integer knapsack polytope
|
V
|
. This shows that the bin packing problem can be solved efficiently when the underlying integer knapsack polytope has an easy structure (i.e., has a small number of vertices). Furthermore, we show that the presented bounds of the structure theorem are asymptotically tight. We give a construction of bin packing instances using new structural insights and classical number theoretical theorems which yield the desired lower bound.</description><identifier>ISSN: 0364-765X</identifier><identifier>EISSN: 1526-5471</identifier><identifier>DOI: 10.1287/moor.2019.1040</identifier><language>eng</language><publisher>Linthicum: INFORMS</publisher><subject>90C60 ; Algorithms ; Apexes ; Asymptotic methods ; bin packing ; Integer programming ; Integers ; Lower bounds ; Mathematical functions ; Operations research ; Packing problem ; Polytopes ; Primary: mathematics ; secondary: sets: polyhedra ; structural properties ; Theorems</subject><ispartof>Mathematics of operations research, 2020-11, Vol.45 (4), p.1498-1511</ispartof><rights>Copyright Institute for Operations Research and the Management Sciences Nov 2020</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><citedby>FETCH-LOGICAL-c368t-b9546f3ea0fff537fef4c5f3ef15da10835e465dfafc17e11837e7b363da85013</citedby><cites>FETCH-LOGICAL-c368t-b9546f3ea0fff537fef4c5f3ef15da10835e465dfafc17e11837e7b363da85013</cites><orcidid>0000-0002-0188-9492</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktohtml>$$Uhttps://pubsonline.informs.org/doi/full/10.1287/moor.2019.1040$$EHTML$$P50$$Ginforms$$H</linktohtml><link.rule.ids>314,777,781,3679,27905,27906,62595</link.rule.ids></links><search><creatorcontrib>Jansen, Klaus</creatorcontrib><creatorcontrib>Klein, Kim-Manuel</creatorcontrib><title>About the Structure of the Integer Cone and Its Application to Bin Packing</title><title>Mathematics of operations research</title><description>We consider the bin packing problem with
d
different item sizes and revisit the structure theorem given by Goemans and Rothvoß about solutions of the integer cone. We present new techniques on how solutions can be modified and give a new structure theorem that relies on the set of vertices of the underlying integer polytope. As a result of our new structure theorem, we obtain an algorithm for the bin packing problem with running time
|
V
|
2
O
(
d
)
⋅
e
n
c
(
I
)
O
(
1
)
, where
V
is the set of vertices of the integer knapsack polytope, and
e
n
c
(
I
)
is the encoding length of the bin packing instance. The algorithm is fixed-parameter tractable, parameterized by the number of vertices of the integer knapsack polytope
|
V
|
. This shows that the bin packing problem can be solved efficiently when the underlying integer knapsack polytope has an easy structure (i.e., has a small number of vertices). Furthermore, we show that the presented bounds of the structure theorem are asymptotically tight. We give a construction of bin packing instances using new structural insights and classical number theoretical theorems which yield the desired lower bound.</description><subject>90C60</subject><subject>Algorithms</subject><subject>Apexes</subject><subject>Asymptotic methods</subject><subject>bin packing</subject><subject>Integer programming</subject><subject>Integers</subject><subject>Lower bounds</subject><subject>Mathematical functions</subject><subject>Operations research</subject><subject>Packing problem</subject><subject>Polytopes</subject><subject>Primary: mathematics</subject><subject>secondary: sets: polyhedra</subject><subject>structural properties</subject><subject>Theorems</subject><issn>0364-765X</issn><issn>1526-5471</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2020</creationdate><recordtype>article</recordtype><recordid>eNqFkE1LxDAYhIMouK5ePQcEb61J89U91sWPlQUFFbyFbJvUrrtJTVLQf29qvXt6YXhmhncAOMcox0UprvbO-bxAeJFjRNEBmGFW8IxRgQ_BDBFOM8HZ2zE4CWGLEGYC0xl4qDZuiDC-a_gc_VDHwWvozK-wslG32sOlsxoq28BVDLDq-11Xq9g5C6OD152FT6r-6Gx7Co6M2gV99nfn4PX25mV5n60f71bLap3VhJcx2ywY5YZohYwxjAijDa1ZEgxmjcKoJExTzhqjTI2FxrgkQosN4aRRJUOYzMHFlNt79znoEOXWDd6mSllQnjoKWopEXU5Uq3ZadrZ26Zuv2KohBCkrTmnKonSRwHwCa-9C8NrI3nd75b8lRnIcVo7DynFYOQ6bDNlk6Kxxfh_-438AigJ5eg</recordid><startdate>20201101</startdate><enddate>20201101</enddate><creator>Jansen, Klaus</creator><creator>Klein, Kim-Manuel</creator><general>INFORMS</general><general>Institute for Operations Research and the Management Sciences</general><scope>AAYXX</scope><scope>CITATION</scope><scope>JQ2</scope><orcidid>https://orcid.org/0000-0002-0188-9492</orcidid></search><sort><creationdate>20201101</creationdate><title>About the Structure of the Integer Cone and Its Application to Bin Packing</title><author>Jansen, Klaus ; Klein, Kim-Manuel</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c368t-b9546f3ea0fff537fef4c5f3ef15da10835e465dfafc17e11837e7b363da85013</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2020</creationdate><topic>90C60</topic><topic>Algorithms</topic><topic>Apexes</topic><topic>Asymptotic methods</topic><topic>bin packing</topic><topic>Integer programming</topic><topic>Integers</topic><topic>Lower bounds</topic><topic>Mathematical functions</topic><topic>Operations research</topic><topic>Packing problem</topic><topic>Polytopes</topic><topic>Primary: mathematics</topic><topic>secondary: sets: polyhedra</topic><topic>structural properties</topic><topic>Theorems</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Jansen, Klaus</creatorcontrib><creatorcontrib>Klein, Kim-Manuel</creatorcontrib><collection>CrossRef</collection><collection>ProQuest Computer Science Collection</collection><jtitle>Mathematics of operations research</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Jansen, Klaus</au><au>Klein, Kim-Manuel</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>About the Structure of the Integer Cone and Its Application to Bin Packing</atitle><jtitle>Mathematics of operations research</jtitle><date>2020-11-01</date><risdate>2020</risdate><volume>45</volume><issue>4</issue><spage>1498</spage><epage>1511</epage><pages>1498-1511</pages><issn>0364-765X</issn><eissn>1526-5471</eissn><abstract>We consider the bin packing problem with
d
different item sizes and revisit the structure theorem given by Goemans and Rothvoß about solutions of the integer cone. We present new techniques on how solutions can be modified and give a new structure theorem that relies on the set of vertices of the underlying integer polytope. As a result of our new structure theorem, we obtain an algorithm for the bin packing problem with running time
|
V
|
2
O
(
d
)
⋅
e
n
c
(
I
)
O
(
1
)
, where
V
is the set of vertices of the integer knapsack polytope, and
e
n
c
(
I
)
is the encoding length of the bin packing instance. The algorithm is fixed-parameter tractable, parameterized by the number of vertices of the integer knapsack polytope
|
V
|
. This shows that the bin packing problem can be solved efficiently when the underlying integer knapsack polytope has an easy structure (i.e., has a small number of vertices). Furthermore, we show that the presented bounds of the structure theorem are asymptotically tight. We give a construction of bin packing instances using new structural insights and classical number theoretical theorems which yield the desired lower bound.</abstract><cop>Linthicum</cop><pub>INFORMS</pub><doi>10.1287/moor.2019.1040</doi><tpages>14</tpages><orcidid>https://orcid.org/0000-0002-0188-9492</orcidid></addata></record> |
| fulltext | fulltext |
| identifier | ISSN: 0364-765X |
| ispartof | Mathematics of operations research, 2020-11, Vol.45 (4), p.1498-1511 |
| issn | 0364-765X 1526-5471 |
| language | eng |
| recordid | cdi_proquest_journals_2463682487 |
| source | Informs |
| subjects | 90C60 Algorithms Apexes Asymptotic methods bin packing Integer programming Integers Lower bounds Mathematical functions Operations research Packing problem Polytopes Primary: mathematics secondary: sets: polyhedra structural properties Theorems |
| title | About the Structure of the Integer Cone and Its Application to Bin Packing |
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