On the Complexity of Robust Multi-Stage Problems in the Polynomial Hierarchy

TL;DR

This paper investigates the computational complexity of multi-stage robust optimization problems within the polynomial hierarchy, revealing how problem type and uncertainty discretization influence their hardness levels.

Contribution

It introduces a technique to prove $ ext{Sigma}^p_3$-hardness for robust multi-stage problems and distinguishes complexity differences between continuous and discrete uncertainty.

Findings

Discrete uncertainty leads to $ ext{Sigma}^p_3$-completeness for several problems.

Continuous uncertainty generally keeps problems in the first stage of the hierarchy.

Allowing uncertainty in multiple constraints raises complexity to the third stage even in continuous cases.

Abstract

We study the computational complexity of multi-stage robust optimization problems. Such problems are formulated with alternating min/max quantifiers and therefore naturally fall into a higher stage of the polynomial hierarchy. Despite this, almost no hardness results with respect to the polynomial hierarchy are known. In this work, we examine the hardness of robust two-stage adjustable and robust recoverable optimization with budgeted uncertainty sets. Our main technical contribution is the introduction of a technique tailored to prove $\Sigma^p_3$-hardness of such problems. We highlight a difference between continuous and discrete budgeted uncertainty: In the discrete case, indeed a wide range of problems becomes complete for the third stage of the polynomial hierarchy; in particular, this applies to the TSP, independent set, and vertex cover problems. However, in the continuous case this does not happen and problems remain in the first stage of the hierarchy. Finally, if we allow the uncertainty to not only affect the objective, but also multiple constraints, then this distinction disappears and even in the continuous case we encounter hardness for the third stage of the hierarchy. This shows that even robust problems which are already NP-complete can still exhibit a significant computational difference between column-wise and row-wise uncertainty.