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# Cutting Planes and the Elementary Closure in Fixed Dimension

1 PROTHEO - Constraints, automatic deduction and software properties proofs
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-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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https://hal.inria.fr/inria-00098944
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Submitted on : Tuesday, September 26, 2006 - 8:40:38 AM
Last modification on : Friday, February 4, 2022 - 3:31:28 AM

### Identifiers

• HAL Id : inria-00098944, version 1

### Citation

Alexander Bockmayr, Friedrich Eisenbrand. Cutting Planes and the Elementary Closure in Fixed Dimension. [Intern report] 99-R-362 || bockmayr_99a, 1999, 12 p. ⟨inria-00098944⟩

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