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Complexity of linear relaxations in integer programming.

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Summary
This summary is machine-generated.

The relaxation complexity measures the complexity of integer points within polyhedra. This study advances understanding of this complexity, providing new bounds and computational results for various conditions.

Keywords:
03C1052B0552B20Primary: 90C10Secondary: 90C57

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Area of Science:

  • Discrete Geometry
  • Integer Programming
  • Computational Geometry

Background:

  • The relaxation complexity, introduced by Kaibel & Weltge (2015), quantifies the minimum facets needed for a polyhedron to represent a specific set of integer points.
  • This parameter is crucial for understanding the complexity of linear descriptions of integer point sets without auxiliary variables.

Purpose of the Study:

  • To address open questions concerning the relaxation complexity and its variant for rational polyhedra.
  • To establish new bounds and computational methods for determining the relaxation complexity.

Main Methods:

  • Utilizing tools from combinatorics and the geometry of numbers.
  • Applying techniques from quantifier elimination.
  • Analyzing properties of integer point sets within polyhedra, including dimension, residue class representation, convex hull properties, and lattice-width.

Main Results:

  • Established that the relaxation complexity is bounded for integer point sets in at most four dimensions.
  • Proved bounds for sets representing all residue classes or containing an interior integer point.
  • Demonstrated that relaxation complexity is algorithmically computable under specific dimensional and property-based conditions.
  • Derived an improved lower bound for relaxation complexity based on the dimension of the integer point set.

Conclusions:

  • Significant progress has been made in characterizing and computing the relaxation complexity of integer point sets.
  • The findings provide a deeper understanding of the complexity of integer descriptions in polyhedra.
  • New conditions and computational approaches enhance the practical applicability of relaxation complexity in related fields.