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The '''Karmarkar-Karp (KK) 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.
The KK algorithms were 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 name=":0">{{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> The KK algorithms were the first ones to attain an additive approximation.
The KK 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.
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**''n<sub>i</sub>'' is the number of items of size ''s<sub>i</sub>''.
* ''B'' - the bin size.
* ''e'' - a fraction in (0,1), such that ''eB'' is the smallest size of an item.▼
== High-level idea ==
The KK algorithms essentially solve the [[configuration linear program]]:<blockquote><math>\text{minimize}~~\mathbf{1}\cdot \mathbf{x}~~~\text{s.t.}~~ A \mathbf{x}\geq \mathbf{n}~~~\text{and}~~ \mathbf{x}\geq 0~~~\text{and}~~ \mathbf{x}~\text{is an integer}~</math>.</blockquote>Here, '''''A''''' is a matrix with ''m'' rows. Each column of '''''A''''' represents a feasible ''configuration'' - a multiset of item-sizes, such that the sum of all these sizes is at most ''B''. The set of configurations is ''C''. '''x''' is a vector of size ''C.'' Each element ''x<sub>c</sub>'' of '''x''' represents the number of times configuration ''c'' is used.
* ''Example'':<ref name=":22">{{Cite web|last=Claire Mathieu|title=Approximation Algorithms Part I, Week 3: bin packing|url=https://www.coursera.org/learn/approximation-algorithms-part-1/home/week/3|url-status=live|website=Coursera}}</ref> suppose the item sizes are 3,3,3,3,3,4,4,4,4,4, and ''B''=12. Then there are ''C''=10 possible configurations: 3333; 333; 33, 334; 3, 34, 344; 4, 44, 444. The martix '''''A''''' has two rows: [4,3,2,2,1,1,1,0,0,0] for s=3 and [0,0,0,1,0,1,2,1,2,3] for s=4. The vector '''n''' is [5,5] since there are 5 items of each size. A possible optimal solution is '''x'''=[1,0,0,0,0,0,1,0,0,1], corresponding to using three bins with configurations 3333, 344, 444.
There are two main difficulties in solving this problem. First, it is an [[Integer programming|integer linear program]], which is computationally hard to solve. Second, the number of variables is ''C'' - the number of configurations, which may be enormous. The KK algorithms cope with these difficulties using several techniques, some of which were already introduced by de-la-Vega and Lueker. The high-level description of the algorithm is:<ref name=":0" />
* <math>I</math>
== Guarantees ==
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* Adding to this the rounded-down bins of the principal part yields LOPT(I) + S/2.
* The execution time of this conversion algorithm is O(''n'' log ''n'').
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== References ==
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