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

2026-02-18

AI Summary

This study investigates the computational complexity of two-stage robust selection problems under budget uncertainty, focusing on the case where the nominal problem is the simple selection problem. By integrating techniques from computational complexity theory, combinatorial optimization, and robust optimization, the paper systematically analyzes three variants of the selection problem under three types of budget uncertainty sets—covering both discrete and continuous formulations—through reductions and algorithmic constructions. The main contributions include the first complete characterization of the NP-hardness of the two-stage selection problem under continuous budget uncertainty, a proof that the representative selection problem is solvable in polynomial time, and consequent complexity results for the assignment problem, thereby resolving a long-standing open question in the field.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Constraint Satisfaction and Optimization: Mixed Discrete/Continuous Optimization
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Economics, Online Markets and Human Computation: Research challenges in human and human-AI computation
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Security and Privacy: Large-scale security measurements

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.

Problem

• two-stage robust optimization
• budgeted uncertainty
• selection problem
• computational complexity
• NP-hardness

Innovation

• two-stage robust optimization
• budgeted uncertainty
• selection problem
• computational complexity
• NP-hardness