Global tolerances in the problems of combinatorial optimization with an additive objective function

Abstract

It is known that by means of minimal values of tolerances one can obtain necessary and sufficient conditions for the uniqueness of the optimal solution of a combinatorial optimization problem (COP) with an additive objective function and the set of nonembedded feasible solutions. Moreover, the notion of a tolerance is defined locally, i.e., with respect to a chosen optimal solution. In this paper we introduce the notion of a global tolerance with respect to the whole set of optimal solutions and prove that the nonembeddedness assumption on the set of feasible solutions of the COP can be relaxed, which generalizes the well known relations for the extremal values of the tolerances. In particular, we formulate a new criterion for the uniqueness of the optimal solution of the COP with an additive objective function, which is based on certain equalities between locally and globally defined tolerances.