Recoverable Robust Representatives Selection Problems with Discrete Budgeted Uncertainty

TL;DR

This paper studies a recoverable robust selection problem under discrete budgeted uncertainty, establishing its NP-hardness, identifying polynomial cases, and proposing efficient mixed-integer programming formulations with computational comparisons.

Contribution

It proves the NP-hardness of the problem, identifies a polynomial solvable case, and develops multiple mixed-integer programming formulations for effective solution.

Findings

The problem is NP-hard in general.

A special polynomial-time solvable case is identified.

Extended formulations improve computational efficiency.

Abstract

Recoverable robust optimization is a multi-stage approach, where it is possible to adjust a first-stage solution after the uncertain cost scenario is revealed. We analyze this approach for a class of selection problems. The aim is to choose a fixed number of items from several disjoint sets, such that the worst-case costs after taking a recovery action are as small as possible. The uncertainty is modeled as a discrete budgeted set, where the adversary can increase the costs of a fixed number of items. While special cases of this problem have been studied before, its complexity has remained open. In this work we make several contributions towards closing this gap. We show that the problem is NP-hard and identify a special case that remains solvable in polynomial time. We provide a compact mixed-integer programming formulation and two additional extended formulations. Finally, computational results are provided that compare the efficiency of different exact solution approaches.