Extending the definition of set tolerances

TL;DR

This paper broadens the definition of set tolerances in combinatorial optimization to encompass all elements and subsets, maintaining key theoretical and computational properties.

Contribution

It extends the existing definitions of set tolerances to all elements and subsets, ensuring broader applicability while preserving important properties.

Findings

Extended the definition of tolerances to all elements and subsets.

Proved that key properties still hold under the new definitions.

Maintained computational feasibility of the extended tolerances.

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.