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On the Unlimited Number of Faces in Integer Hulls of Linear Programs with a Single Constraint

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  • David S. Rubin

    (University of Chicago, Chicago, Illinois)

Abstract

The convex hull of the feasible integer points to a given integer program is a convex polytope I . The feasible set obtained by relaxing the integrality requirements is another convex polytope L . Cutting-plane algorithms essentially try to remove part of L − I . Hence the more complicated the relationship between L and I , the more difficult (in some sense) the integer program. This paper shows one such complexity: specifically, we construct a series of programs such that I has arbitrarily many faces even though L is a triangle. We also indicate the existence of a large class of problems that exhibit the same behavior.

Suggested Citation

  • David S. Rubin, 1970. "On the Unlimited Number of Faces in Integer Hulls of Linear Programs with a Single Constraint," Operations Research, INFORMS, vol. 18(5), pages 940-946, October.
    • Handle: RePEc:inm:oropre:v:18:y:1970:i:5:p:940-946
    DOI: 10.1287/opre.18.5.940
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    Cited by:

    1. S. Thomas McCormick & Scott R. Smallwood & Frits C. R. Spieksma, 2001. "A Polynomial Algorithm for Multiprocessor Scheduling with Two Job Lengths," Mathematics of Operations Research, INFORMS, vol. 26(1), pages 31-49, February.
    2. Imre Barany & Roger Howe & Laszlo Lovasz, 1989. "On Integer Points in Polyhedra: A Lower Bound," Cowles Foundation Discussion Papers 917, Cowles Foundation for Research in Economics, Yale University.

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