Computation of Set Tolerances with Applications to the Minimum Spanning Tree Problem

TL;DR

This paper develops linear programming methods to compute regular set tolerances in combinatorial sum problems, including the Minimum Spanning Tree Problem, enhancing sensitivity analysis techniques.

Contribution

It introduces novel linear programming approaches and recursive procedures for calculating set tolerances, along with new bounds and exact formulas for small subsets.

Findings

Linear programming approaches for upper and lower tolerances

Recursive procedures for all subset tolerances

Exact formulas for sets of size 2 and 3

Abstract

The regular set tolerance is an important term in sensitivity analysis. For combinatorial sum problems, e.g., the Traveling Salesman Problem, Shortest Path Problem and Minimum Spanning Tree Problem, it determines how much the sum of the costs of the elements of a set can be increased while ensuring that all current optimal solutions remain optimal. The regular set lower tolerance determines how much the sum of the costs of the elements of a set can be decreased while ensuring that the objective value of the optimal solution is not changed. We investigate a general method for computing regular (upper and lower) set tolerances in combinatorial sum problems. For the upper tolerance, we present a linear programming approach, and for the lower tolerances, three linear programming approaches, where the last two are novel and lead to recursive procedures for computation of the lower tolerances of all subsets of the given ground set. Furthermore, we give new upper bounds for set lower tolerances. For both upper and lower tolerances, we give an exact formula for sets of cardinality 2 and 3. Finally, we consider the computation of tolerances for the Minimum Spanning Tree Problem, give a formula for single tolerances, a lower bound for regular set upper tolerances and an exact formula for regular set lower tolerances.