Completeness for the Complexity Class $\forall \exists \mathbb{R}$ and Area-Universality

Abstract: Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class $\exists \mathbb{R}$ plays a crucial role in the study of geometric problems. Sometimes $\exists \mathbb{R}$ is referred to as the 'real analog' of NP. While NP is a class of computational problems that deals with existentially quantified boolean variables, $\exists \mathbb{R}$ deals with existentially quantified real variables. In analogy to $\Pi_2<sup>p$ and $\Sigma_2<sup>p$ in the famous polynomial hierarchy, we study the complexity classes $\forall \exists \mathbb{R}$ and $\exists \forall \mathbb{R}$ with real variables. Our main interest is the area-universality problem, where we are given a plane graph $G$, and ask if for each assignment of areas to the inner faces of $G$, there exists a straight-line drawing of $G$ realizing the assigned areas. We conjecture that area-universality is $\forall \exists \mathbb{R}$-complete and support this conjecture by proving $\exists \mathbb{R}$- and $\forall \exists \mathbb{R}$-completeness of two variants of area-universality. To this end, we introduce tools to prove $\forall \exists \mathbb{R}$-hardness and membership. Finally, we present geometric problems as candidates for $\forall \exists \mathbb{R}$-complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability.