Integer programming: Difference between revisions

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* 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|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.|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}}</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}}</ref> presented a differentan improved algorithm. Thewith improved runtime isrun-time <math>n^{2.5 n} \cdot 2^{O(n)} \ell</math>cdot ,(m\cdot \log where <math>\ellV)^{O(1)}</math> is the number of input bits,.<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> which is in <math>\text{poly}(m, \log{V})</math>.<ref>{{Cite journal|last1=Bredereck|first1=Robert|last2=Kaczmarczyk|first2=Andrzej|last3=Knop|first3=Dušan|last4=Niedermeier|first4=Rolf|date=2019-06-17|title=High-Multiplicity Fair Allocation: Lenstra Empowered by N-fold Integer Programming|url=https://doi.org/10.1145/3328526.3329649|journal=Proceedings of the 2019 ACM Conference on Economics and Computation|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>.

===Heuristic methods===