Completeness for the Complexity Class [Formula: see text] and Area-Universality

Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class [Formula: see text] plays a crucial role in the study of geometric problems. Sometimes [Formula: see text] is referred to as the ‘real analog’ of NP. While NP is a class of computational problems that deals with existentially quantified boolean variables, [Formula: see text] deals with existentially quantified real variables. In analogy to [Formula: see text] and [Formula: see text] in the famous polynomial hierarchy, we study the complexity classes [Formula: see text] and [Formula: see text] 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 [Formula: see text] -complete and support this conjecture by proving [Formula: see text] - and [Formula: see text] -completeness of two variants of Area Universality. To this end, we introduce tools to prove [Formula: see text] -hardness and membership. Finally, we present geometric problems as candidates for [Formula: see text] -complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability.