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^p and Sigma _2^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 AreaUniversality 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 AreaUniversality 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 AreaUniversality. 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.