Karmarkar–Karp bin packing algorithms: Difference between revisions

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The '''Karmarkar-Karp bin packing algorithms''' are several related [[approximation algorithm]] for the [[bin packing problem]].<ref name=":12">{{cite journal|last1=Karmarkar|first1=Narendra|last2=Karp|first2=Richard M.|date=November 1982|title=An efficient approximation scheme for the one-dimensional bin-packing problem|url=https://ieeexplore.ieee.org/document/4568405/references#references|journal=23rd Annual Symposium on Foundations of Computer Science (SFCS 1982)|pages=312–320|doi=10.1109/SFCS.1982.61|s2cid=18583908}}</ref> The bin packing problem is a problem of packing items of different sizes into bins of identical capacity, such that the total number of bins is as small as possible. Finding the optimal solution is [[NP-hardness|computationally hard]]. [[Narendra Karmarkar|Karmarkar]] and [[Richard M. Karp|Karp]] devised an algorithm that runs in [[Polynomial-time|polynomial time]] and finds a solution with at most <math>\mathrm{OPT} + \mathcal{O}(\log^2(OPT))</math> bins, where OPT is the number of bins in the optimal solution. They also devised several other algorithms with slightly different approximation guarantees and run-time bounds.

Their algorithmalgorithms waswere considered a breakthrough in the study of bin packing: the previously-known algorithms found multiplicative approximation, where the number of bins was at most <math>r\cdot \mathrm{OPT}+s</math> for some constants <math>r>1, s>0</math>, or at most <math>(1+\varepsilon)\mathrm{OPT} + 1</math>.<ref>{{cite journal|last1=Fernandez de la Vega|first1=W.|last2=Lueker|first2=G. S.|date=1981|title=Bin packing can be solved within 1 + ε in linear time|journal=Combinatorica|language=en|volume=1|issue=4|pages=349–355|doi=10.1007/BF02579456|issn=1439-6912|s2cid=10519631}}</ref> Their techniques were improved later, to provide even better approximations: an algorithm by Rothvoss<ref name=":2">{{Cite journal|last=Rothvoß|first=T.|date=2013-10-01|title=Approximating Bin Packing within O(log OPT * Log Log OPT) Bins|url=https://ieeexplore.ieee.org/document/6686137|journal=2013 IEEE 54th Annual Symposium on Foundations of Computer Science|volume=|pages=20–29|arxiv=1301.4010|doi=10.1109/FOCS.2013.11|isbn=978-0-7695-5135-7|via=|s2cid=15905063}}</ref> uses at most <math>\mathrm{OPT} + O(\log(\mathrm{OPT})\cdot \log\log(\mathrm{OPT}))</math>bins, and an algorithm by Hoberg and Rothvoss<ref name=":3">{{Citation|last1=Hoberg|first1=Rebecca|title=A Logarithmic Additive Integrality Gap for Bin Packing|date=2017-01-01|url=https://epubs.siam.org/doi/abs/10.1137/1.9781611974782.172|work=Proceedings of the 2017 Annual ACM-SIAM Symposium on Discrete Algorithms|pages=2616–2625|series=Proceedings|publisher=Society for Industrial and Applied Mathematics|doi=10.1137/1.9781611974782.172|isbn=978-1-61197-478-2|access-date=2021-02-10|last2=Rothvoss|first2=Thomas|s2cid=1647463}}</ref> uses at most <math>\mathrm{OPT} + O(\log(\mathrm{OPT}))</math> bins.

== Input ==

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* ''B'' - the bin size.

* ''e'' - a fraction in (0,1), such that ''eB'' is the smallest size of an item.

* ''FOPT'' = (''a''₁+...+''a_n'')/''B'' = the theoretically-optimal number of bins, when all bins are completely filled with items or item fractions.

== Guarantees ==

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* At most <math>(1+\epsilon)\mathrm{OPT} + \mathcal{O}(\epsilon^{-2})</math> bins, with run-time in <math>O(T(\epsilon^{-2},n))</math>, where <math>\epsilon>0</math> is a constant.

== AlgorithmsFractional bin packing ==

* ''FOPT'' = (''a''₁+...+''a_n'')/''B'' = the theoretically-optimal number of bins, when all bins are completely filled with items or item fractions.

*

== References ==