Assessing the effect of multiple cost changes using reverse set tolerances

We determine the sensitivity of a current optimal solution to a combinatorial optimization problem to cost changes in a set of elements. In a recent study, the concept of regular set tolerances has been introduced for a combinatorial optimization problem and for three types of cost functions, namely sum, product, and bottleneck. A regular set tolerance is the supremum amount of cost changes that can be distributed in the most favorable way to multiple elements such as not to change the current optimal solution. In this paper, we introduce an alternative concept, namely the reverse set tolerance, which is a measure of the infimum amount of cost changes to multiple elements such that the current optimal solution becomes non-optimal. We characterize the specific cases in which reverse set upper and lower tolerances have positive values and in which they are infinite. We also show a criterion for the uniqueness of an optimal solution. Furthermore, we present bounds and exact formulas for reverse set upper and lower tolerances using the relation to their corresponding single tolerance counterparts. We discuss the similarities and differences in the results between regular and reverse set tolerances. Finally, we motivate this new concept by analyzing them for special combinatorial optimization problems with important practical applications.