On the representativeness of approximate solutions of discrete optimization problems with interval objective function

TL;DR

This paper investigates the representativeness of approximate solutions in discrete optimization problems with interval uncertainty, showing that mean and other solutions can often be unrepresentative even with small errors.

Contribution

It introduces probabilistic notions of possible and approximate solutions under interval uncertainty and analyzes their representativeness in discrete optimization.

Findings

Mean approximate solutions can be unrepresentative for small errors.

All possible approximate solutions may be unrepresentative.

Unrepresentativeness can be significant even with small boundary values.

Abstract

We consider discrete optimization problems with interval uncertatinty of objective function coefficients. The interval uncertainty models measurements errors. A pos\-sible optimal solution is a solution that is optimal for some possible values of the coefficients. Pro\-ba\-bi\-li\-ty of a possible solution is the probability to obtain such coefficients that the solution is optimal. Similarly we define the notion of a possible approximate solution with given accuracy and probability of the solution. A possible approximate solution is an approximate solution that is obtained for some possible values of the coefficients by some fixed approximate algorithm, e.g. by the greedy algorithm. Pro\-ba\-bi\-li\-ty of a such solution is the probability to obtain such coefficients that the algorithm produces the solution as its output. We consider optimal or approximate possible solution un\-re\-pre\-sen\-ta\-ti\-ve if its probability less than some boundary value. The mean approximate solution is a possible approximate solution for midpoints of the coefficients intervals. The solution may be treated as approximate solution for exact values of the coefficients. We show that the share of individual discrete optimization problems instances with unrepresentative mean approximate solution may be wide enough for rather small values of error and the boundary value. The same is true for any other possible approximate solution: all of them may be unrepresentative.