Completeness for the Complexity Class ∀∃R and Area-Universality.

Abstract

Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class $\exists R$ plays a crucial role in the study of geometric problems. Sometimes $\exists R$ is referred to as the 'real analog' of NP. While NP is a class of computational problems that deals with existentially quantified boolean variables, $\exists R$ deals with existentially quantified real variables. In analogy to ${\Pi }_2^p$ and ${\Sigma }_2^p$ in the famous polynomial hierarchy, we study the complexity classes $\forall \exists R$ and $\exists \forall R$ 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 $\forall \exists R$-complete and support this conjecture by proving $\exists R$- and $\forall \exists R$-completeness of two variants of Area Universality. To this end, we introduce tools to prove $\forall \exists R$-hardness and membership. Finally, we present geometric problems as candidates for $\forall \exists R$-complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability.