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 < 0}. Computational experience shows that C i has “few” generators. We demonstrate this in the first nontrivial case n = 3, m = 4.
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© 2008 Herbert Scarf
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Bárány, I., Scarf, H. (2008). Matrices with Identical Sets of Neighbors. In: Yang, Z. (eds) Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research. Palgrave Macmillan, London. https://doi.org/10.1057/9781137024411_10
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DOI: https://doi.org/10.1057/9781137024411_10
Publisher Name: Palgrave Macmillan, London
Print ISBN: 978-1-137-02440-4
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