Content deleted Content added
→Further reading: add INRIA overview article |
m Open access bot: url-access=subscription updated in citation with #oabot. |
||
| (2 intermediate revisions by the same user not shown) | |||
Line 102:
* [[Cashflow matching|Cash flow matching]]
* [[Energy system]] optimization<ref>{{Cite journal|last1=Morais|first1=Hugo|last2=Kádár|first2=Péter|last3=Faria|first3=Pedro|last4=Vale|first4=Zita A.|last5=Khodr|first5=H. M.|date=2010-01-01|title=Optimal scheduling of a renewable micro-grid in an isolated load area using mixed-integer linear programming|url=http://www.sciencedirect.com/science/article/pii/S0960148109001001|journal=Renewable Energy|language=en|volume=35|issue=1|pages=151–156|doi=10.1016/j.renene.2009.02.031|bibcode=2010REne...35..151M |issn=0960-1481|hdl=10400.22/1585|hdl-access=free|url-access=subscription}}</ref><ref>{{Cite journal|last1=Omu|first1=Akomeno|last2=Choudhary|first2=Ruchi|last3=Boies|first3=Adam|date=2013-10-01|title=Distributed energy resource system optimisation using mixed integer linear programming|url=http://www.sciencedirect.com/science/article/pii/S0301421513003418|journal=Energy Policy|language=en|volume=61|pages=249–266|doi=10.1016/j.enpol.2013.05.009|bibcode=2013EnPol..61..249O |s2cid=29369795 |issn=0301-4215|url-access=subscription}}</ref>
* [[Unmanned aerial vehicle|UAV]] [[Guidance system|guidance]]<ref>{{Cite book|last1=Schouwenaars|first1=T.|last2=Valenti|first2=M.|last3=Feron|first3=E.|last4=How|first4=J.|title=2005 IEEE Aerospace Conference |chapter=Implementation and Flight Test Results of MILP-based UAV Guidance |date=2005|pages=1–13|doi=10.1109/AERO.2005.1559600|isbn=0-7803-8870-4|s2cid=13447718}}</ref><ref>{{Cite journal|last1=Radmanesh|first1=Mohammadreza|last2=Kumar|first2=Manish|date=2016-03-01|title=Flight formation of UAVs in presence of moving obstacles using fast-dynamic mixed integer linear programming|journal=Aerospace Science and Technology|language=en|volume=50|pages=149–160|doi=10.1016/j.ast.2015.12.021|issn=1270-9638|doi-access=free|bibcode=2016AeST...50..149R }}</ref>
* [[Transit map]] [[Graph drawing|layouting]]<ref>{{cite arXiv |last1=Bast |first1=Hannah |last2=Brosi |first2=Patrick |last3=Storandt |first3=Sabine |date=2017-10-05 |title=Efficient Generation of Geographically Accurate Transit Maps |class=cs.CG |eprint=1710.02226 }}</ref>
Line 135:
Suppose <math>A</math> is an ''m''-by-''n'' integer matrix and <math>\mathbf{b}</math> is an ''m''-by-1 integer vector. We focus on the feasibility problem, which is to decide whether there exists an ''n''-by-1 vector <math>\mathbf{x}</math> satisfying <math> A \mathbf{x} \le \mathbf{b} </math>.
Let ''V'' be the maximum absolute value of the coefficients in <math>A</math> and <math>\mathbf{b}</math>. If ''n'' (the number of variables) is a fixed constant, then the feasibility problem can be solved in time polynomial in ''m'' and log ''V''. This is trivial for the case ''n''=1. The case ''n''=2 was solved in 1981 by [[Herbert Scarf]].<ref>{{Cite journal|last=Scarf|first=Herbert E.|date=1981|title=Production Sets with Indivisibilities, Part I: Generalities|url=https://www.jstor.org/stable/1911124|journal=Econometrica|volume=49|issue=1|pages=1–32|doi=10.2307/1911124|jstor=1911124|issn=0012-9682|url-access=subscription}}</ref> The general case was solved in 1983 by [[Hendrik Lenstra]], combining ideas by [[László Lovász]] and [[Peter van Emde Boas]].<ref name=":0">{{Cite journal|last=Lenstra|first=H. W.|date=1983-11-01|title=Integer Programming with a Fixed Number of Variables|url=https://pubsonline.informs.org/doi/abs/10.1287/moor.8.4.538|journal=Mathematics of Operations Research|volume=8|issue=4|pages=538–548|doi=10.1287/moor.8.4.538|issn=0364-765X|citeseerx=10.1.1.431.5444}}</ref> [[Doignon's theorem]] asserts that an integer program is feasible whenever every subset of <math>2^n</math> constraints is feasible; a method combining this result with algorithms for [[LP-type problem]]s can be used to solve integer programs in time that is linear in <math>m</math> and [[fixed-parameter tractable]] (FPT) in ''<math>n</math>'', but possibly [[Double exponential function|doubly exponential]] in <math>n</math>, with no dependence on <math>V</math>.<ref>{{cite conference
| last1 = Amenta | first1 = Nina | author1-link = Nina Amenta
| last2 = De Loera | first2 = Jesús A. | author2-link = Jesús A. De Loera
Line 157:
* The original algorithm of Lenstra<ref name=":0" /> had run-time <math>2^{O(n^3)}\cdot (m\cdot \log V)^{O(1)}</math>.
* Kannan<ref>{{Cite journal|last=Kannan|first=Ravi|date=1987-08-01|title=Minkowski's Convex Body Theorem and Integer Programming|url=https://pubsonline.informs.org/doi/abs/10.1287/moor.12.3.415|journal=Mathematics of Operations Research|volume=12|issue=3|pages=415–440|doi=10.1287/moor.12.3.415|s2cid=495512 |issn=0364-765X}}</ref> presented an improved algorithm with run-time <math>n^{O(n)}\cdot (m\cdot \log V)^{O(1)}</math>.<ref>{{Cite journal|last1=Goemans|first1=Michel X.|author1link = Michel Goemans|last2=Rothvoss|first2=Thomas|date=2020-11-07|title=Polynomiality for Bin Packing with a Constant Number of Item Types|journal=[[Journal of the ACM]]|volume=67|issue=6|pages=38:1–38:21|doi=10.1145/3421750|hdl=1721.1/92865 |s2cid=227154747 |issn=0004-5411|doi-access=free|hdl-access=free|arxiv=1307.5108}}</ref>
* Frank and Tardos<ref>{{Cite journal|last1=Frank|first1=András|last2=Tardos|first2=Éva|date=1987-03-01|title=An application of simultaneous diophantine approximation in combinatorial optimization|url=https://doi.org/10.1007/BF02579200|journal=Combinatorica|language=en|volume=7|issue=1|pages=49–65|doi=10.1007/BF02579200|s2cid=45585308|issn=1439-6912|url-access=subscription}}</ref> presented an improved algorithm with run-time <math>n^{2.5 n} \cdot 2^{O(n)} \cdot (m\cdot \log V)^{O(1)}</math>.<ref>{{Cite journal|last1=Bliem|first1=Bernhard|last2=Bredereck|first2=Robert|last3=Niedermeier|first3=Rolf|author3-link=Rolf Niedermeier|date=2016-07-09|title=Complexity of efficient and envy-free resource allocation: few agents, resources, or utility levels|url=https://dl.acm.org/doi/abs/10.5555/3060621.3060636|journal=Proceedings of the Twenty-Fifth International Joint Conference on Artificial Intelligence|series=IJCAI'16|___location=New York, New York, USA|publisher=AAAI Press|pages=102–108|isbn=978-1-57735-770-4}}</ref><ref>{{Cite book|last1=Bredereck|first1=Robert|last2=Kaczmarczyk|first2=Andrzej|last3=Knop|first3=Dušan|last4=Niedermeier|first4=Rolf|title=Proceedings of the 2019 ACM Conference on Economics and Computation |chapter=High-Multiplicity Fair Allocation: Lenstra Empowered by N-fold Integer Programming |date=2019-06-17|chapter-url=https://doi.org/10.1145/3328526.3329649|series=EC '19|___location=Phoenix, AZ, USA|publisher=Association for Computing Machinery|pages=505–523|doi=10.1145/3328526.3329649|isbn=978-1-4503-6792-9|s2cid=195298520}}</ref>{{Rp|Prop.8}}
* Dadush<ref>Dadush, Daniel (2012-06-14). [https://homepages.cwi.nl/~dadush/papers/dadush-thesis.pdf "Integer Programming, Lattice Algorithms, and Deterministic Volume Estimation].</ref> presented an improved algorithm with run-time <math>n^n \cdot 2^{O(n)} \cdot (m \cdot \log V)^{O(1)}</math>.
* Reis and Rothvoss<ref>Reis, Victor; Rothvoss, Thomas (2023-03-26). [https://arxiv.org/abs/2303.14605 "The Subspace Flatness Conjecture and Faster Integer Programming"].</ref> presented an improved algorithm with run-time <math>(\log n)^{O(n)} \cdot (m\cdot \log V)^{O(1)}</math>.
| |||