Applications and efficient algorithms for integer programming problems on monotone constraints
We present here classes of integer programming problems that are solvable efficiently and with combinatorial flow algorithms. The problems are characterized by constraints that have either at most two variables per inequality that appear with opposite sign coefficients, or have in addition a third v...
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| Vydáno v: | Networks Ročník 77; číslo 1; s. 21 - 49 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Hoboken, USA
John Wiley & Sons, Inc
01.01.2021
Wiley Subscription Services, Inc |
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| ISSN: | 0028-3045, 1097-0037 |
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| Abstract | We present here classes of integer programming problems that are solvable efficiently and with combinatorial flow algorithms. The problems are characterized by constraints that have either at most two variables per inequality that appear with opposite sign coefficients, or have in addition a third variable that appears only in one constraint. Such integer programs, referred to here as monotone IP2 or IP3, are shown to be solvable in polynomial time for polynomially bounded variables. This article demonstrates the vast applicability of IP2 and IP3 as models for integer programs in multiple scenarios. Since the problems are easily recognized, the knowledge of their structure enables one to determine easily that they are efficiently solvable. The variety of applications, that previously were not known to be solved as efficiently, underlies the importance of recognizing this structure and, if appropriate, formulating problems as monotone IP2 or IP3. Additionally, if there is flexibility in the modeling of an integer programming problem, the formulation choice as monotone IP2 or IP3 leads to efficient algorithms, whereas slightly different modeling choices would lead to NP‐hard problems. |
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| AbstractList | We present here classes of integer programming problems that are solvable efficiently and with combinatorial flow algorithms. The problems are characterized by constraints that have either at most two variables per inequality that appear with opposite sign coefficients, or have in addition a third variable that appears only in one constraint. Such integer programs, referred to here as monotone IP2 or IP3, are shown to be solvable in polynomial time for polynomially bounded variables. This article demonstrates the vast applicability of IP2 and IP3 as models for integer programs in multiple scenarios. Since the problems are easily recognized, the knowledge of their structure enables one to determine easily that they are efficiently solvable. The variety of applications, that previously were not known to be solved as efficiently, underlies the importance of recognizing this structure and, if appropriate, formulating problems as monotone IP2 or IP3. Additionally, if there is flexibility in the modeling of an integer programming problem, the formulation choice as monotone IP2 or IP3 leads to efficient algorithms, whereas slightly different modeling choices would lead to NP‐hard problems. |
| Author | Hochbaum, Dorit S. |
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| Cites_doi | 10.1287/mnsc.22.11.1268 10.1137/0214016 10.1109/CVPR.2006.91 10.1109/34.868688 10.1523/ENEURO.0304-18.2019 10.1007/s10479-009-0643-z 10.1287/ijoc.1120.0546 10.1137/0218003 10.1109/ICCV.2009.5459261 10.1007/s00453-004-1085-2 10.1093/bioinformatics/bts232 10.1016/S0377-2217(02)00071-1 10.1016/j.ejor.2018.07.009 10.1109/ICPR.1996.546886 10.1145/502090.502093 10.4208/nmtma.2013.mssvm09 10.1007/s10479-019-03464-z 10.1287/opre.1120.1126 10.1109/TPAMI.2009.80 10.1007/3-540-69346-7_25 10.1007/s13675-012-0002-8 10.1109/34.969114 10.1038/nature04977 10.1287/opre.1080.0524 10.1137/S0097539793251876 10.4153/CJM-1956-045-5 |
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| SubjectTerms | Algorithms Combinatorial analysis image segmentation Integer programming Modelling monotone inequalities network flow parametric cut Polynomials ratio problems |
| Title | Applications and efficient algorithms for integer programming problems on monotone constraints |
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