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The Frobenius Problem and Maximal Lattice Free Bodies

In: Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research

Author

Listed:
  • Herbert E. Scarf

    (Yale University)

  • David F. Shallcross

    (Bell Communications Research)

Abstract

Let p = (p1,…,pn,) be a vector of positive integers whose greatest common divisor is unity. The Frobenius problem is to find the largest integer f* which cannot be written as a nonnegative integral combination of the pi. In this note we relate the Frobenius problem to the topic of maximal lattice free bodies and describe an algorithm for n = 3.

Suggested Citation

  • Herbert E. Scarf & David F. Shallcross, 2008. "The Frobenius Problem and Maximal Lattice Free Bodies," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 7, pages 149-153, Palgrave Macmillan.
    • Handle: RePEc:pal:palchp:978-1-137-02441-1_7
    DOI: 10.1057/9781137024411_7
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    References listed on IDEAS

    as
    1. H. W. Lenstra, 1983. "Integer Programming with a Fixed Number of Variables," Mathematics of Operations Research, INFORMS, vol. 8(4), pages 538-548, November.
    2. Herbert E. Scarf, 2008. "Production Sets with Indivisibilities Part II. The Case of Two Activities," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 3, pages 39-67, Palgrave Macmillan.
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    Cited by:

    1. Herbert E. Scarf & Kevin M. Woods, 2008. "Neighborhood Complexes and Generating Functions for Affine Semigroups," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 12, pages 207-225, Palgrave Macmillan.

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