Polynomiality for Bin Packing with a Constant Number of Item Types
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Goemans_Polynomiality for.pdf
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Author(s)
Goemans, Michel X.
Rothvoss, Thomas
Date Issued
2014
Journal
Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms
Publisher
Society for Industrial and Applied Mathematics
Citation
Goemans, Michel X., and Thomas Rothvoß. “Polynomiality for Bin Packing with a Constant Number of Item Types.” Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms (December 18, 2013): 830–839.
Version
Final published version
Abstract
We consider the bin packing problem with d different item sizes si and item multiplicities ai, where all numbers are given in binary encoding. This problem formulation is also known as the 1-dimensional cutting stock problem.
In this work, we provide an algorithm which, for constant d, solves bin packing in polynomial time. This was an open problem for all d ≥ 3.
In fact, for constant d our algorithm solves the following problem in polynomial time: given two d-dimensional polytopes P and Q, find the smallest number of integer points in P whose sum lies in Q.
Our approach also applies to high multiplicity scheduling problems in which the number of copies of each job type is given in binary encoding and each type comes with certain parameters such as release dates, processing times and deadlines. We show that a variety of high multiplicity scheduling problems can be solved in polynomial time if the number of job types is constant.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1137/1.9781611973402.61
