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

Abstract

Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class ∃R plays a crucial role in the study of geometric problems. Sometimes ∃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, ∃R deals with existentially quantified real variables. In analogy to Πp2 and Σp2 in the famous polynomial hierarchy, we study the complexity classes ∀∃R and ∃∀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 ∀∃R -complete and support this conjecture by proving ∃R - and ∀∃R -completeness of two variants of AREA UNIVERSALITY. To this end, we introduce tools to prove ∀∃R -hardness and membership. Finally, we present geometric problems as candidates for ∀∃R -complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability.