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Cutting planes and the elementary closure in fixed dimension

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http://pubman.mpdl.mpg.de/cone/persons/resource/persons44149

Bockmayr,  Alexander
Programming Logics, MPI for Informatics, Max Planck Society;

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1999-2-008
(Any fulltext), 10KB

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Citation

Bockmayr, A.(1999). Cutting planes and the elementary closure in fixed dimension (MPI-I-1999-2-008). Saarbrücken: Max-Planck-Institut für Informatik.


Cite as: http://hdl.handle.net/11858/00-001M-0000-0014-6F4F-C
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 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 $Q_I$.