# Cutting Planes and the Elementary Closure in Fixed Dimension

1 MODBIO - Computational models in molecular biology
INRIA Lorraine, LORIA - Laboratoire Lorrain de Recherche en Informatique et ses Applications
Abstract : The elementary closure $P'$ of a polyhedron $P$ is the intersection of $P$ with all its Gomory-Chv{á}tal 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 simplicial cone, we construct a polytope $Q$, whose integral elements correspond to cutting planes of $P$. The vertices of the integer hull $Q_I$ 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 $Q_I$.
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Type de document :
Article dans une revue
Mathematics of Operations Research, INFORMS, 2001, 26 (2), pp.304-312
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https://hal.inria.fr/inria-00101099
Contributeur : Publications Loria <>
Soumis le : mardi 26 septembre 2006 - 14:56:30
Dernière modification le : jeudi 11 janvier 2018 - 06:19:51

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• HAL Id : inria-00101099, version 1

### Citation

Alexander Bockmayr, Friedrich Eisenbrand. Cutting Planes and the Elementary Closure in Fixed Dimension. Mathematics of Operations Research, INFORMS, 2001, 26 (2), pp.304-312. 〈inria-00101099〉

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