The Complexity of Zadeh's Pivot Rule
The Simplex algorithm is one of the most important algorithms in discrete optimization, and is the most used algorithm for solving linear programs in practice. In the last 50 years, several pivot rules for this algorithm have been proposed and studied. For most deterministic pivot rules, exponential...
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| Formato: | Online |
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| Idioma: | inglês |
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Logos Verlag Berlin
2022
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| Acesso em linha: | OCN: 1235833310 |
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| collection | Directory of Open Access Books |
| description | The Simplex algorithm is one of the most important algorithms in discrete optimization, and is the most used algorithm for solving linear programs in practice. In the last 50 years, several pivot rules for this algorithm have been proposed and studied. For most deterministic pivot rules, exponential lower bounds were found, while a probabilistic pivot rule exists that guarantees termination in expected subexponential time.
One deterministic pivot rule that is of special interest is Zadeh's pivot rule since it was the most promising candidate for a polynomial pivot rule for a long time. In 2011, Friedmann proved that this is not true by providing an example forcing the Simplex algorithm to perform at least a subexponential number of iterations in the worst case when using Zadeh's pivot rule. Still, it was not known whether Zadeh's pivot rule might achieve subexponential worst case running time. Next to analyzing Friedmann's construction in detail, we develop the first exponential lower bound for Zadeh's pivot rule. This closes a long-standing open problem by ruling out this pivot rule as a candidate for a deterministic, subexponential pivot rule in several areas of linear optimization and game theory. |
| format | Online |
| id | doab-20.500.12854ir-84219 |
| institution | Directory of Open Access Books |
| language | eng |
| publishDate | 2022 |
| publishDateRange | 2022 |
| publishDateSort | 2022 |
| publisher | Logos Verlag Berlin |
| publisherStr | Logos Verlag Berlin |
| record_format | ojs |
| spelling | doab-20.500.12854ir-842192025-07-30T07:20:00Z The Complexity of Zadeh's Pivot Rule Hopp, Alexander Vincent Computers Computer Science The Simplex algorithm is one of the most important algorithms in discrete optimization, and is the most used algorithm for solving linear programs in practice. In the last 50 years, several pivot rules for this algorithm have been proposed and studied. For most deterministic pivot rules, exponential lower bounds were found, while a probabilistic pivot rule exists that guarantees termination in expected subexponential time. One deterministic pivot rule that is of special interest is Zadeh's pivot rule since it was the most promising candidate for a polynomial pivot rule for a long time. In 2011, Friedmann proved that this is not true by providing an example forcing the Simplex algorithm to perform at least a subexponential number of iterations in the worst case when using Zadeh's pivot rule. Still, it was not known whether Zadeh's pivot rule might achieve subexponential worst case running time. Next to analyzing Friedmann's construction in detail, we develop the first exponential lower bound for Zadeh's pivot rule. This closes a long-standing open problem by ruling out this pivot rule as a candidate for a deterministic, subexponential pivot rule in several areas of linear optimization and game theory. 2022-06-19T04:01:38Z 2022-06-19T04:01:38Z 2022-06-18T05:38:39Z 2020 book OCN: 1235833310 https://library.oapen.org/handle/20.500.12657/56796 9783832552060 https://directory.doabooks.org/handle/20.500.12854/84219 eng open access image/jpeg image/jpeg image/jpeg image/jpeg image/jpeg image/jpeg n/a n/a n/a n/a n/a n/a https://library.oapen.org/bitstream/20.500.12657/56796/1/external_content.pdf https://library.oapen.org/bitstream/20.500.12657/56796/1/external_content.pdf https://library.oapen.org/bitstream/20.500.12657/56796/1/external_content.pdf https://library.oapen.org/bitstream/20.500.12657/56796/1/external_content.pdf https://library.oapen.org/bitstream/20.500.12657/56796/1/external_content.pdf https://library.oapen.org/bitstream/20.500.12657/56796/1/external_content.pdf Logos Verlag Berlin Logos Verlag Berlin https://doi.org/10.30819/5206 https://doi.org/10.30819/5206 04b263a1-7fba-4491-9eae-1c394ac42fc3 Knowledge Unlatched 9783832552060 Knowledge Unlatched (KU) KU Open Services Logos Verlag Berlin open access |
| spellingShingle | Computers Computer Science The Complexity of Zadeh's Pivot Rule |
| title | The Complexity of Zadeh's Pivot Rule |
| title_full | The Complexity of Zadeh's Pivot Rule |
| title_fullStr | The Complexity of Zadeh's Pivot Rule |
| title_full_unstemmed | The Complexity of Zadeh's Pivot Rule |
| title_short | The Complexity of Zadeh's Pivot Rule |
| title_sort | complexity of zadeh s pivot rule |
| topic | Computers Computer Science |
| topic_facet | Computers Computer Science |
| url | OCN: 1235833310 |