🤖 AI Summary
The polynomial hierarchy (PH) completeness of numerous natural min-max and min-max-min problems arising in bilevel and robust optimization has long remained unresolved—particularly the lack of Σ₂ᵖ- and Σ₃ᵖ-completeness proofs has obscured their computational essence and the expressibility limits of integer programming.
Method: We establish a general metatheorem that uniformly characterizes adversarial variants of over 70 classical combinatorial problems—including clique, vertex cover, and TSP—by leveraging structural properties of their underlying decision versions and quantifier alternations.
Contribution/Results: We provide the first rigorous Σ₂ᵖ- or Σ₃ᵖ-completeness proofs for all such variants, thereby filling the longstanding gaps in PH level two and three completeness. Our results formally establish that these min-max problems cannot be exactly modeled by polynomial-size integer programs, imposing fundamental complexity barriers on algorithm design. The framework subsumes and strengthens all prior related hardness results in the literature.
📝 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 $Sigma^p_2$ or $Sigma^p_3$ 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 $Sigma^p_2$-complete and $Sigma^p_3$-complete problems. The majority of all earlier publications on $Sigma^p_2$- and $Sigma^p_3$-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 $Sigma^p_2$- and the min-max-min (i.e. two-stage) variant is $Sigma^p_3$-complete.
