Geometric Algorithms and Combinatorial Optimization:
Gespeichert in:
| 1. Verfasser: | |
|---|---|
| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Berlin, Heidelberg
Springer Berlin Heidelberg
1988
|
| Schriftenreihe: | Algorithms and Combinatorics
2 |
| Schlagworte: | |
| Online-Zugang: | Volltext |
| Beschreibung: | Historically, there is a close connection between geometry and optimization. This is illustrated by methods like the gradient method and the simplex method, which are associated with clear geometric pictures. In combinatorial optimization, however, many of the strongest and most frequently used algorithms are based on the discrete structure of the problems: the greedy algorithm, shortest path and alternating path methods, branch-and-bound, etc. In the last several years geometric methods, in particular polyhedral combinatorics, have played a more and more profound role in combinatorial optimization as well. Our book discusses two recent geometric algorithms that have turned out to have particularly interesting consequences in combinatorial optimization, at least from a theoretical point of view. These algorithms are able to utilize the rich body of results in polyhedral combinatorics. The first of these algorithms is the ellipsoid method, developed for nonlinear programming by N. Z. Shor, D. B. Yudin, and A. S. NemirovskiI. It was a great surprise when L. G. Khachiyan showed that this method can be adapted to solve linear programs in polynomial time, thus solving an important open theoretical problem. While the ellipsoid method has not proved to be competitive with the simplex method in practice, it does have some features which make it particularly suited for the purposes of combinatorial optimization. The second algorithm we discuss finds its roots in the classical "geometry of numbers", developed by Minkowski. This method has had traditionally deep applications in number theory, in particular in diophantine approximation |
| Beschreibung: | 1 Online-Ressource (XII, 362p. 23 illus) |
| ISBN: | 9783642978814 9783642978838 |
| ISSN: | 0937-5511 |
| DOI: | 10.1007/978-3-642-97881-4 |
Internformat
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Datensatz im Suchindex
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| author | Grötschel, Martin 1948- |
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| building | Verbundindex |
| bvnumber | BV042423162 |
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| dewey-full | 511.6 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 511 - General principles of mathematics |
| dewey-raw | 511.6 |
| dewey-search | 511.6 |
| dewey-sort | 3511.6 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1007/978-3-642-97881-4 |
| format | Electronic eBook |
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| id | DE-604.BV042423162 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T01:21:12Z |
| institution | BVB |
| isbn | 9783642978814 9783642978838 |
| issn | 0937-5511 |
| language | English |
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| series | Algorithms and Combinatorics |
| series2 | Algorithms and Combinatorics |
| spelling | Grötschel, Martin 1948- Verfasser (DE-588)108975282 aut Geometric Algorithms and Combinatorial Optimization by Martin Grötschel, László Lovász, Alexander Schrijver Berlin, Heidelberg Springer Berlin Heidelberg 1988 1 Online-Ressource (XII, 362p. 23 illus) txt rdacontent c rdamedia cr rdacarrier Algorithms and Combinatorics 2 0937-5511 Historically, there is a close connection between geometry and optimization. This is illustrated by methods like the gradient method and the simplex method, which are associated with clear geometric pictures. In combinatorial optimization, however, many of the strongest and most frequently used algorithms are based on the discrete structure of the problems: the greedy algorithm, shortest path and alternating path methods, branch-and-bound, etc. In the last several years geometric methods, in particular polyhedral combinatorics, have played a more and more profound role in combinatorial optimization as well. Our book discusses two recent geometric algorithms that have turned out to have particularly interesting consequences in combinatorial optimization, at least from a theoretical point of view. These algorithms are able to utilize the rich body of results in polyhedral combinatorics. The first of these algorithms is the ellipsoid method, developed for nonlinear programming by N. Z. Shor, D. B. Yudin, and A. S. NemirovskiI. It was a great surprise when L. G. Khachiyan showed that this method can be adapted to solve linear programs in polynomial time, thus solving an important open theoretical problem. While the ellipsoid method has not proved to be competitive with the simplex method in practice, it does have some features which make it particularly suited for the purposes of combinatorial optimization. The second algorithm we discuss finds its roots in the classical "geometry of numbers", developed by Minkowski. This method has had traditionally deep applications in number theory, in particular in diophantine approximation Mathematics Combinatorics Mathematik Kombinatorische Optimierung (DE-588)4031826-6 gnd rswk-swf Geometrie (DE-588)4020236-7 gnd rswk-swf Kombinatorische Geometrie (DE-588)4140733-7 gnd rswk-swf Polyedrische Kombinatorik (DE-588)4132100-5 gnd rswk-swf Algorithmische Geometrie (DE-588)4130267-9 gnd rswk-swf Polynomialzeitalgorithmus (DE-588)4199652-5 gnd rswk-swf Kombinatorische Optimierung (DE-588)4031826-6 s Polyedrische Kombinatorik (DE-588)4132100-5 s Polynomialzeitalgorithmus (DE-588)4199652-5 s 1\p DE-604 Geometrie (DE-588)4020236-7 s 2\p DE-604 Algorithmische Geometrie (DE-588)4130267-9 s 3\p DE-604 Kombinatorische Geometrie (DE-588)4140733-7 s 4\p DE-604 Lovász, László 1948- Sonstige (DE-588)108767337 oth Schrijver, Alexander 1948- Sonstige (DE-588)13576355X oth Algorithms and Combinatorics 2 (DE-604)BV000617357 2 https://doi.org/10.1007/978-3-642-97881-4 Verlag Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 2\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 3\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk 4\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Grötschel, Martin 1948- Geometric Algorithms and Combinatorial Optimization Algorithms and Combinatorics Mathematics Combinatorics Mathematik Kombinatorische Optimierung (DE-588)4031826-6 gnd Geometrie (DE-588)4020236-7 gnd Kombinatorische Geometrie (DE-588)4140733-7 gnd Polyedrische Kombinatorik (DE-588)4132100-5 gnd Algorithmische Geometrie (DE-588)4130267-9 gnd Polynomialzeitalgorithmus (DE-588)4199652-5 gnd |
| subject_GND | (DE-588)4031826-6 (DE-588)4020236-7 (DE-588)4140733-7 (DE-588)4132100-5 (DE-588)4130267-9 (DE-588)4199652-5 |
| title | Geometric Algorithms and Combinatorial Optimization |
| title_auth | Geometric Algorithms and Combinatorial Optimization |
| title_exact_search | Geometric Algorithms and Combinatorial Optimization |
| title_full | Geometric Algorithms and Combinatorial Optimization by Martin Grötschel, László Lovász, Alexander Schrijver |
| title_fullStr | Geometric Algorithms and Combinatorial Optimization by Martin Grötschel, László Lovász, Alexander Schrijver |
| title_full_unstemmed | Geometric Algorithms and Combinatorial Optimization by Martin Grötschel, László Lovász, Alexander Schrijver |
| title_short | Geometric Algorithms and Combinatorial Optimization |
| title_sort | geometric algorithms and combinatorial optimization |
| topic | Mathematics Combinatorics Mathematik Kombinatorische Optimierung (DE-588)4031826-6 gnd Geometrie (DE-588)4020236-7 gnd Kombinatorische Geometrie (DE-588)4140733-7 gnd Polyedrische Kombinatorik (DE-588)4132100-5 gnd Algorithmische Geometrie (DE-588)4130267-9 gnd Polynomialzeitalgorithmus (DE-588)4199652-5 gnd |
| topic_facet | Mathematics Combinatorics Mathematik Kombinatorische Optimierung Geometrie Kombinatorische Geometrie Polyedrische Kombinatorik Algorithmische Geometrie Polynomialzeitalgorithmus |
| url | https://doi.org/10.1007/978-3-642-97881-4 |
| volume_link | (DE-604)BV000617357 |
| work_keys_str_mv | AT grotschelmartin geometricalgorithmsandcombinatorialoptimization AT lovaszlaszlo geometricalgorithmsandcombinatorialoptimization AT schrijveralexander geometricalgorithmsandcombinatorialoptimization |