Recoverable Robust Representatives Selection Problems with Discrete Budgeted Uncertainty

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.

Combinatorial Optimization Problems with Balanced Regret

“…For multi-stage problems there can be a difference in complexity depending on whether continuous or budgeted uncertainty is used. For example, the two-stage selection problem with continuous budgeted uncertainty can be solved in polynomial time [CGKZ18], but becomes NP-hard for discrete budgeted uncertainty [GLW20]. Observe that this is not the case for balanced regret.…”

Two-Stage Robust Optimization Problems with Two-Stage Uncertainty

“…We now consider budgeted uncertainty sets as defined in Section 2 with the difference that the adversarial variables δ i determining the distribution of the uncertainty budget need to be discrete, i.e., we have δ i ∈ {0, 1} instead of δ i ∈ [0, 1]. While for one-stage problems, this does not have any impact on the problem, it is well known to make a difference for two-stage problems (see, e.g., [CGKZ18,GLW20]). Discrete variables in the inner adversarial recourse problem AdvRec(x x x, c c c, y y y) can be relaxed without changing the optimal objective value, which means that we find the same recourse problem Rec(x x x, c c c) as before in (2).…”