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MPI-INF/SWS Research Reports 1991-2017

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MPI-I-1999-2-008

Cutting planes and the elementary closure in fixed dimension

Bockmayr, Alexander and Eisenbrand, Friedrich

December 1999, 12 pages.

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Status: available - back from printing

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 $Q_I$.

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URL to this document: http://domino.mpi-inf.mpg.de/internet/reports.nsf/NumberView/1999-2-008

Hide details for BibTeXBibTeX
@TECHREPORT{BockmayrEisenbrandMPI-I-1999-2-008,
  AUTHOR = {Bockmayr, Alexander and Eisenbrand, Friedrich},
  TITLE = {Cutting planes and the elementary closure in fixed dimension},
  TYPE = {Research Report},
  INSTITUTION = {Max-Planck-Institut f{\"u}r Informatik},
  ADDRESS = {Stuhlsatzenhausweg 85, 66123 Saarbr{\"u}cken, Germany},
  NUMBER = {MPI-I-1999-2-008},
  MONTH = {December},
  YEAR = {1999},
  ISSN = {0946-011X},
}