Gomory-Chvátal Cutting Planes and the Elementary Closure of Polyhedra
نویسندگان
چکیده
The elementary closure P ′ of a polyhedron P is the intersection of P with all its GomoryChvátal cutting planes. P ′ is a rational polyhedron provided that P is rational. The Chvátal-Gomory procedure is the iterative application of the elementary closure operation to P . The Chvátal rank is the minimal number of iterations needed to obtain PI . It is always finite, but already in R2 one can construct polytopes of arbitrary large Chvátal rank. We show that the Chvátal rank of polytopes contained in the n-dimensional 0/1 cube is O(n2 log n) and prove the lower bound (1 + ǫ)n, for some ǫ > 0. We show that the separation problem for the elementary closure of a rational polyhedron is NP-hard. This solves a problem posed by Schrijver. Last we consider the elementary closure in fixed dimension. 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. Finally, we present a polynomial algorithm in varying dimension, which computes cutting planes for a simplicial cone from this polynomial description in fixed dimension with a maximal degree of violation in a natural sense.
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