The complexity landscape of robust (integer) linear programming

Abstract

We study the computational complexity of the decision versions of three classic robust optimization problems: static robust optimization, two-stage (adjustable) robust optimization, and $K$-adaptability. We consider that the feasibility, uncertainty, and recourse sets are polyhedra or integer-programming-representable sets, the uncertainty is decision-independent or decision-dependent, and the feasible regions are bounded or unbounded. While these problems are well-established in the current literature, we give a systematic classification that places the resulting problems in $\mathsf{P}$, $\mathsf{NP}$, $\mathsf{coNP}$, and higher levels of the polynomial hierarchy ($\Sigma_2^p$ and $\Sigma_3^p$), and, once decision-dependent uncertainty introduces quadratic constraints, in the existential theory of the reals and its hierarchy ($\exists\mathbb{R}$, $\Sigma_2\mathbb{R}$, and $\Sigma_3\mathbb{R}$) or among the undecidable problems. Beyond hardness reductions, we pay particular attention to membership proofs, establishing polynomial-size certificates even though the sets considered generally contain vectors of exponential encoding length. As a by-product, we give an alternative proof that bilevel linear optimization lies in $\mathsf{NP}$, exploiting its connection to decision-dependent robust optimization. For $K$-adaptability, we relate the problem to its static and two-stage counterparts, showing that hardness grows monotonically with $K$ and that, for fully discrete decision-dependent instances, $K$-adaptability reduces back to the static problem.