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Matrices with Identical Sets of Neighbors

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

Author

Listed:
  • Imre Bárány

    (Mathematical Institute of the Hungarian Academy of Sciences)

  • Herbert Scarf

    (Yale University)

Abstract

Given a generic m by n matrix A, a lattice point h in ℤn is a neighbor of the origin if the body {x: Ax ≤ b}, with b1 = max {0, aih }, i = 1, …,m, contains no lattice point other than 0 and h. The set of neighbors, N(A), is finite and 0-symmetric. We show that if A′ is another matrix of the same size with the property that sign a i h = sign a′ i h for every i and every h ∈ N(A), then A′ has precisely the same set of neighbors as A. The collection of such matrices is a polyhedral cone, described by a finite set of linear inequalities, each such inequality corresponding to a generator of one of the cones C i = pos {h ∈ N(A): a i h

Suggested Citation

  • Imre Bárány & Herbert Scarf, 2008. "Matrices with Identical Sets of Neighbors," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 10, pages 179-189, Palgrave Macmillan.
    • Handle: RePEc:pal:palchp:978-1-137-02441-1_10
    DOI: 10.1057/9781137024411_10
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    References listed on IDEAS

    as
    1. Imre Bárány & Roger Howe & Herbert E. Scarf, 2008. "The complex of maximal lattice free simplices," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 8, pages 155-163, Palgrave Macmillan.
    2. Herbert E. Scarf, 2008. "Neighborhood Systems for Production Sets with Indivisibilities," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 5, pages 105-130, Palgrave Macmillan.
    3. I. Bárány & H. E. Scarf & D. Shallcross, 2008. "The topological structure of maximal lattice free convex bodies: The general case," Palgrave Macmillan Books, in: Zaifu Yang (ed.), Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, chapter 11, pages 191-205, Palgrave Macmillan.
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