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Extending the definition of set tolerances

Gerold Jäger, Marcel Turkensteen

TL;DR

The definition of single and set tolerances is broadened to all elements for single tolerances and to all subsets of elements for set tolerances, while proving that key theoretical and computational properties still apply.

Abstract

Optimal solutions of combinatorial optimization problems can be sensitive to changes in the cost of one or more elements of the ground set E. Single and set tolerances measure the supremum / infimum possible change such that the current solution remains optimal for cost changes in one or more elements. The current definition does not apply to all elements of E or to all subsets of E. In this work, we broaden the definition to all elements for single tolerances and to all subsets of elements for set tolerances, while proving that key theoretical and computational properties still apply.

Extending the definition of set tolerances

TL;DR

The definition of single and set tolerances is broadened to all elements for single tolerances and to all subsets of elements for set tolerances, while proving that key theoretical and computational properties still apply.

Abstract

Optimal solutions of combinatorial optimization problems can be sensitive to changes in the cost of one or more elements of the ground set E. Single and set tolerances measure the supremum / infimum possible change such that the current solution remains optimal for cost changes in one or more elements. The current definition does not apply to all elements of E or to all subsets of E. In this work, we broaden the definition to all elements for single tolerances and to all subsets of elements for set tolerances, while proving that key theoretical and computational properties still apply.
Paper Structure (13 sections, 10 equations)

This paper contains 13 sections, 10 equations.

Table of Contents

  1. Introduction
  2. Notations, definitions, and existing results
  3. Combinatorial minimization problems
  4. Single upper tolerances
  5. Single lower tolerances
  6. Regular and reverse set upper tolerances
  7. Regular and reverse set lower tolerances
  8. Interpretation of regular and reverse set tolerances
  9. Examples
  10. Generalized definitions of upper and lower tolerances
  11. New and extended definitions of tolerances
  12. Concepts and motivation
  13. Computation, bounds, and properties of the extended upper tolerance definitions

Theorems & Definitions (6)

  • Example 1
  • Example 2
  • Example 3
  • Example 4
  • Remark 1
  • Remark 2