Integer programming with binary and bounded variables via Gröbner bases with applications to multiobjective integer programming
Families of integer programming problems can be solved efficiently in practice once their reduced Gröbner basis is known. However, computing Gröbner bases is often hard, especially when binary or integer bounded variables are present in the problem formulation. In this paper, we study the specific s...
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| Veröffentlicht in: | Journal of symbolic computation Jg. 135; S. 102529 |
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| Abstract | Families of integer programming problems can be solved efficiently in practice once their reduced Gröbner basis is known. However, computing Gröbner bases is often hard, especially when binary or integer bounded variables are present in the problem formulation. In this paper, we study the specific structure of the constraint matrix of integer programs with bounded variables and the implications of this structure to the truncated Gröbner bases of integer programming problems. In this direction, we introduce a new Binary Truncation Criterion that is capable of predicting and eliminating useless S-vectors before they built in the Gröbner basis computation. Additionally, we propose improvements to the Gröbner basis approach to multiobjective integer programming of Jiménez-Tafur (2017) and Hartillo-Hermoso et al. (2020), such as a proof that truncated Gröbner bases can be used in their algorithm with no loss of correctness, implying that our new truncation techniques are also useful in this application. All new proposed methods are implemented in the open source package IPGBs and their performance is empirically validated. |
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| AbstractList | Families of integer programming problems can be solved efficiently in practice once their reduced Gröbner basis is known. However, computing Gröbner bases is often hard, especially when binary or integer bounded variables are present in the problem formulation. In this paper, we study the specific structure of the constraint matrix of integer programs with bounded variables and the implications of this structure to the truncated Gröbner bases of integer programming problems. In this direction, we introduce a new Binary Truncation Criterion that is capable of predicting and eliminating useless S-vectors before they built in the Gröbner basis computation. Additionally, we propose improvements to the Gröbner basis approach to multiobjective integer programming of Jiménez-Tafur (2017) and Hartillo-Hermoso et al. (2020), such as a proof that truncated Gröbner bases can be used in their algorithm with no loss of correctness, implying that our new truncation techniques are also useful in this application. All new proposed methods are implemented in the open source package IPGBs and their performance is empirically validated. |
| ArticleNumber | 102529 |
| Author | Mattos Langeloh, Gabriel |
| Author_xml | – sequence: 1 givenname: Gabriel orcidid: 0009-0005-9533-6301 surname: Mattos Langeloh fullname: Mattos Langeloh, Gabriel email: gmlangeloh@inf.ufrgs.br organization: Instituto de Informática, Universidade Federal do Rio Grande do Sul, Porto Alegre, Brazil |
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| Cites_doi | 10.1007/s10107-003-0385-7 10.1287/moor.27.4.681.305 10.1016/S0022-4049(99)00009-2 10.1016/0001-8708(82)90048-2 10.1016/S0927-0507(05)12003-9 10.1287/moor.20.4.864 10.1007/BF01585566 10.1137/S0895480195281209 10.1002/mcda.1780 10.1007/s10107-002-0315-0 10.1016/S0167-6377(03)00098-1 10.2748/tmj/1178227496 10.1016/j.jsc.2003.04.003 10.1016/S0747-7171(88)80048-8 10.4310/MRL.1999.v6.n1.a6 10.1007/s10288-011-0165-9 10.1287/moor.1040.0099 10.1016/j.ejor.2019.09.032 10.1016/S0022-4049(99)00005-5 10.1007/BF02614622 10.1016/j.cor.2013.05.019 10.1007/s002000050062 10.1016/j.jsc.2016.07.031 10.1007/s10589-015-9739-3 |
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