Evaluation of solution of discrete optimization problem with incomplete input

Abstract

We consider the general problem formulation as the minimization of the sum of the numbers of the feasible subset of cardinality n out of the set of N numbers. Asymmetric travelling salesman problem, assignement problem and the problem of minimization of the sum of n numbers out of the set of N numbers can be formulated in these terms. The input of all these problems can be represented as the point from the non-negative part of the unit sphere after some intuitive scaling. In this settings we assume that the retrieval of the input data is difficult. We further propose to skip the retrieval of some input numbers and substitute them with the zeros. We call this problem the problem with incomplete input. We estimate the quality of the optimal solution for the problem with incomplete input used as the solution for the problem with the complete data.