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Hard Equality Constrained Integer Knapsacks

  • Karen Aardal
  • Arjen K. Lenstra
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2337)

Abstract

We consider the following integer feasibility problem: “Given positive integer numbers a 0, a 1,..., a n, with gcd(a 1,..., a n) = 1 and a = (a 1,..., a n), does there exist a nonnegative integer vector x satisfying ax = a 0?” Some instances of this type have been found to be extremely hard to solve by standard methods such as branch-and-bound, even if the number of variables is as small as ten. We observe that not only the sizes of the numbers a 0, a 1,..., a n, but also their structure, have a large impact on the difficulty of the instances. Moreover, we demonstrate that the characteristics that make the instances so difficult to solve by branch-and-bound make the solution of a certain reformulation of the problem almost trivial. We accompany our results by a small computational study.

Keywords

Integer Vector Short Vector Search Node Positive Integer Number Frobenius Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Karen Aardal
    • 1
  • Arjen K. Lenstra
    • 2
    • 3
  1. 1.Mathematisch InstituutUniversiteit UtrechtUtrechtThe Netherlands
  2. 2.Emerging TechnologyCitibank N.A.MendhamUSA
  3. 3.Faculteit Wiskunde en InformaticaTechnische Universiteit EindhovenEindhovenThe Netherlands

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