Non-standard approaches to integer programming

  • Authors:

  • Karen Aardal;Robert Weismantel;Laurence A. Wolsey

  • Affiliations:

  • CORE and INMA, Universite Catholique de Louvain, 34 Voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium and Department of Mathematics, Universiteit Utrecht, Budapestlaan 6, 3584 CH Utrecht, The Net ...;CORE and INMA, Universite Catholique de Louvain, 34 Voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium and Fakultät für Mathematik, IMO Otto-van-Guericke Universität Magdeburg ...;CORE and INMA, Universite Catholique de Louvain, 34 Voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2002

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Abstract

In this survey we address three of the principal algebraic approaches to integer programming. After introducing lattices and basis reduction, we first survey their use in integer programming, presenting among others Lenstra's algorithm that is polynomial in fixed dimension, and the solution of diophanine equations using basis reduction. The second topic concerns augmentation algorithms and test sets, including the role played by Hilbert and Gröbner bases in the development of a primal approach to solve a family of problems for all right-hand sides. Thirdly we survey the group approach of Gomory, showing the importance of subadditivity in integer programming and the generation of valid inequalities, as well the relation to the parametric problem cited above of solving for all right-hand sides.