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Approximation of Corner Polyhedra with Families of Intersection Cuts

  • Gennadiy Averkov
  • Amitabh Basu
  • Joseph PaatEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10328)

Abstract

We study the problem of approximating the corner polyhedron using intersection cuts derived from families of lattice-free sets. In particular, we look at the problem of characterizing families that approximate the corner polyhedron up to a constant factor in fixed dimension n (the constant depends on n). The literature already contains several results in this direction. In this paper, we use the maximum number of facets of a lattice-free set in a family as a measure of its complexity and precisely characterize the level of complexity of a family required for constant factor approximations. As one of the main results, we show that for each natural number n, a corner polyhedron for n integer variables is approximated by intersection cuts from lattice-free sets with at most i facets up to a constant factor (depending only on n) if \(i> 2^{n-1}\) and that no such approximation is possible if \(i \le 2^{n-1}\). When the approximation factor is allowed to depend on the denominator of the underlying fractional point of the corner polyhedron, we show that the threshold is \(i > n\) versus \(i \le n\). The tools introduced for proving such results are of independent interest for studying intersection cuts.

Keywords

Relative Strength Gauge Function Constant Factor Approximation Recession Cone Standard Basis Vector 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Faculty of Mathematics, Institute of Mathematical OptimizationUniversity of MagdeburgMagdeburgGermany
  2. 2.Department of Applied Mathematics and StatisticsJohns Hopkins UniversityBaltimoreUSA

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