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Basis Reduction Methods

  1. Gábor Pataki1,
  2. Mustafa Tural2

Published Online: 14 JAN 2011

DOI: 10.1002/9780470400531.eorms0093

Wiley Encyclopedia of Operations Research and Management Science

Wiley Encyclopedia of Operations Research and Management Science

How to Cite

Pataki, G. and Tural, M. 2011. Basis Reduction Methods. Wiley Encyclopedia of Operations Research and Management Science. .

Author Information

  1. 1

    Department of Statistics and Operations Research, UNC Chapel Hill, Chapel Hill, North Carolina

  2. 2

    Institute for Mathematics and Its Applications, University of Minnesota, Minneapolis, Minnesota

Publication History

  1. Published Online: 14 JAN 2011

Abstract

We review lattice based methods to solve integer programming feasibility problems, in particular, the algorithms of Lenstra, and Kannan, and the reformulation methods of Aardal, et al. and of Krishnamoorthy and Pataki. The unifying theme in all of them is transforming the problem

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where P is a polyhedron, into

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where the columns of B are short, and near orthogonal, that is, they form a reduced basis of the generated lattice, and the choice of B and the polyhedron Q is specific to each method.

We give simple proofs of the polynomial running time of Lenstra's and Kannan's algorithms under the assumption that the dimension is fixed. We analyze the reformulation methods on knapsack problems with decomposable structure, and more surprisingly, we prove that they solve bounded integer programs with high probability by enumerating only one subproblem.

We include several exercises as well, and we believe that the survey will be suitable to teach a 2–3 class long segment on lattice based methods in a course on Integer Programming.

Keywords:

  • reduced bases;
  • Lenstra's algorithm;
  • Kannan's algorithm;
  • lattice-based reformulation methods