Cutting planes and the elementary closure in fixed dimension

Cutting planes and the elementary closure in fixed dimension

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Article ID: iaor20041206
Country: United States
Volume: 26
Issue: 2
Start Page Number: 304
End Page Number: 312
Publication Date: May 2001
Journal: Mathematics of Operations Research
Authors: ,
Keywords: cutting plane algorithms
Abstract:

The elementary closure P′ of a polyhedron P is the intersection of P with all its Gomory–Chvátal cutting planes. P′ is a rational polyhedron provided that P is rational. The known bounds for the number of inequalities defining P′ are exponential, even in fixed dimension. We show that the number of inequalities needed to describe the elementary closure of a rational polyhedron is polynomially bounded in fixed dimension. If P is a simplified cone, we construct a polytope Q, whose integral elements correspond to cutting planes of P. The vertices of the integer hull QI include the facets of P′. A polynomial upper bound on their number can be obtained by applying a result of Cook et al. Finally, we present a polynomial algorithm in varying dimension, which computes cutting planes for a simplicial cone that correspond to vertices of QI.

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