Non-standard approaches to integer programming

K.I. Aardal, R. Weismantel, L.A. Wolsey

    Research output: Contribution to journalArticleAcademicpeer-review

    41 Citations (Scopus)

    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.
    Original languageEnglish
    Pages (from-to)5-74
    JournalDiscrete Applied Mathematics
    Volume123
    Issue number1-3
    DOIs
    Publication statusPublished - 2002

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