The Complexity Landscape of Two-Stage Robust Selection Problems with Budgeted Uncertainty

Abstract

A standard type of uncertainty set in robust optimization is budgeted uncertainty, where an interval of possible values for each parameter is given and the total deviation from their lower bounds is bounded. In the two-stage setting, discrete and continuous budgeted uncertainty have to be distinguished. The complexity of such problems is largely unexplored, in particular if the underlying nominal optimization problem is simple, such as for selection problems. In this paper, we give a comprehensive answer to long-standing open complexity questions for three types of selection problems and three types of budgeted uncertainty sets. In particular, we demonstrate that the two-stage selection problem with continuous budgeted uncertainty is NP-hard, while the corresponding two-stage representative selection problem is solvable in polynomial time. Our hardness result implies that also the two-stage assignment problem with continuous budgeted uncertainty is NP-hard.

Figures (3)

• Figure 1: Example budgeted uncertainty sets.
• Figure 2: Problem \ref{['fjpi"']} can be understood as a continuous min-knapsack problem dependent on parameter $\pi$. We want to pack 1 unit size into the knapsack while minimizing the cost. Index $i$ is associated with two items of size $\pi_i = \min\{1,\pi/d_i\}$ and $1 - \pi_i$, respectively. We depict an example instance with $m=1$ and $\lvert T_1 \rvert=2$ for different values of $\pi$. The order of the four items is given by their ratios $\underline{c}_1 = 1 < \underline{c}_2 = 2 < \overline{c}_2 = 4 < \overline{c}_1 = 5$.
• Figure 3: Illustration of the piecewise linear functions $f_j$ and $f$.

Theorems & Definitions (34)

• Lemma 1
• proof
• Lemma 2
• proof
• Lemma 3
• proof
• Lemma 4
• Theorem 5
• proof
• Lemma 6