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#### Cutting planes and the elementary closure in fixed dimension

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##### Fulltext (public)

1999-2-008

(Any fulltext), 10KB

##### Supplementary Material (public)

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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 $Q_I$.