Gomory-Chvátal Cutting Planes and the Elementary Closure of Polyhedra

نویسندگان

  • Friedrich Eisenbrand
  • Alexander Bockmayr
  • Kurt Mehlhorn
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.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

A Refined Gomory-Chvátal Closure for Polytopes in the Unit Cube

We introduce a natural strengthening of Gomory-Chvátal cutting planes for the important class of 0/1-integer programming problems and study the properties of the elementary closure that arises from the new class of cuts. Most notably, we prove that the new closure is polyhedral, we characterize the family of all facet-defining inequalities, and we compare it to elementary closures associated wi...

متن کامل

Lower Bounds for Chvátal-gomory Style Operators

Valid inequalities or cutting planes for (mixed-) integer programming problems are an essential theoretical tool for studying combinatorial properties of polyhedra. They are also of utmost importance for solving optimization problems in practice; in fact any modern solver relies on several families of cutting planes. The Chvátal-Gomory procedure, one such approach, has a peculiarity that differ...

متن کامل

The Gomory-Chvatal Closure: Polyhedrality, Complexity, and Extensions

In this thesis, we examine theoretical aspects of the Gomory-Chvital closure of polyhedra. A Gomory-Chvital cutting plane for a polyhedron P is derived from any rational inequality that is valid for P by shifting the boundary of the associated half-space towards the polyhedron until it intersects an integer point. The Gomory-ChvAital closure of P is the intersection of all half-spaces defined b...

متن کامل

Corner Polyhedra and Maximal Lattice-free Convex Sets : A Geometric Approach to Cutting Plane Theory

Corner Polyhedra were introduced by Gomory in the early 60s and were studied by Gomory and Johnson. The importance of the corner polyhedron is underscored by the fact that almost all “generic” cutting planes, both in the theoretical literature as well as ones used in practice, are valid for the corner polyhedron. So the corner polyhedron can be viewed as a unifying structure from which many of ...

متن کامل

Origin and early evolution of corner polyhedra

Corner Polyhedra are a natural intermediate step between linear programming and integer programming. This paper first describes how the concept of Corner Polyhedra arose unexpectedly from a practical operations research problem, and then describes how it evolved to shed light on fundamental aspects of integer programming and to provide a great variety of cutting planes for integer programming. ...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2007