Matrices with Identical Sets of Neighbors

Imre Bárány and Herbert Scarf
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Imre Bárány: Mathematical Institute of the Hungarian Academy of Sciences

Chapter 10 in Herbert Scarf’s Contributions to Economics, Game Theory and Operations Research, 2008, pp 179-189 from Palgrave Macmillan

Abstract: 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

Keywords: Test sets for integer programming; polyhedral sets of matrices (search for similar items in EconPapers)
Date: 2008
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Working Paper: Matrices with Identical Sets of Neighbors (1996)
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Persistent link: https://EconPapers.repec.org/RePEc:pal:palchp:978-1-137-02441-1_10

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DOI: 10.1057/9781137024411_10

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