Completeness in the Polynomial Hierarchy for many natural Problems in Bilevel and Robust Optimization

TL;DR

The paper addresses the hardness of min-max and two-stage optimization problems in bilevel and robust settings, arguing that $Σ^p_2$ and $Σ^p_3$ are the natural complexity classes for these decision problems. It introduces a general meta-theorem built on the SSP-NPc framework, showing how to upgrade NP-completeness results to $Σ^p_2$- or $Σ^p_3$-completeness for broad classes of problems via SSP reductions. By identifying a large SSP-NPc class and applying the meta-theorem across network interdiction, min-max regret, and two-stage adjustable optimization, the authors obtain over 70 new $Σ^p_2$/$Σ^p_3$-complete problems and unify prior results. The framework implies that many min-max variants cannot be captured by polynomial-size integer programs, guiding the development of new algorithms and complexity-aware approaches in robust and bilevel optimization.

Abstract

In bilevel and robust optimization we are concerned with combinatorial min-max problems, for example from the areas of min-max regret robust optimization, network interdiction, most vital vertex problems, blocker problems, and two-stage adjustable robust optimization. Even though these areas are well-researched for over two decades and one would naturally expect many (if not most) of the problems occurring in these areas to be complete for the classes $Σ^p_2$ or $Σ^p_3$ from the polynomial hierarchy, almost no hardness results in this regime are currently known. However, such complexity insights are important, since they imply that no polynomial-sized integer program for these min-max problems exist, and hence conventional IP-based approaches fail. We address this lack of knowledge by introducing over 70 new $Σ^p_2$-complete and $Σ^p_3$-complete problems. The majority of all earlier publications on $Σ^p_2$- and $Σ^p_3$-completeness in said areas are special cases of our meta-theorem. Precisely, we introduce a large list of problems for which the meta-theorem is applicable (including clique, vertex cover, knapsack, TSP, facility location and many more). We show that for every single of these problems, the corresponding min-max (i.e. interdiction/regret) variant is $Σ^p_2$- and the min-max-min (i.e. two-stage) variant is $Σ^p_3$-complete.

Figures (1)

• Figure 1: Classic reduction of 3Sat to Vertex Cover for $\varphi = (\overline \ell_1 \lor \overline \ell_2 \lor \ell_3)$.

Theorems & Definitions (1)

• Definition 1: Linear Optimization Problem