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Journal Articles Mathematics of Operations Research Year : 2001

Cutting Planes and the Elementary Closure in Fixed Dimension

Friedrich Eisenbrand
  • Function : Author


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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Dates and versions

inria-00101099 , version 1 (26-09-2006)


  • HAL Id : inria-00101099 , version 1


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