∀∃R-completeness and area-universality
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In the study of geometric problems, the complexity class ∃R turned out to play a crucial role. It exhibits a deep connection between purely geometric problems and real algebra, and is sometimes referred to as the "real analogue" to the class NP. While NP can be considered as a class of computational problems that deals with existentially quantified boolean variables, ∃R deals with existentially quantified real variables. In analogy to Π_2^p and Σ_2^p in the famous polynomial hierarchy, we introduce and motivate the complexity classes ∀∃R and ∃∀R with real variables. Our main interest is focused on 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 is an area-realizing straight-line drawing of G. We conjecture that the problem Area Universality is ∀∃R-complete and support this conjecture by a series of partial results, where we prove ∃R- and ∀∃R-completeness of variants of Area Universality. To do so, we also introduce first tools to study ∀∃R, such as restricted variants of UETR, which are ∀∃R-complete. Finally, we present geometric problems as candidates for ∀∃R-complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability.
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