# $$\forall \exists \mathbb {R}$$-Completeness and Area-Universality

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11159)

## Abstract

In the study of geometric problems, the complexity class $$\exists \mathbb {R}$$ plays a crucial role since it exhibits a deep connection between purely geometric problems and real algebra. Sometimes $$\exists \mathbb {R}$$ is referred to as the “real analogue” to the class 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 $$\varPi _2^p$$ and $$\varSigma _2^p$$ 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 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 $$\forall \exists \mathbb {R}$$-complete and support this conjecture by a series of partial results, where we prove $$\exists \mathbb {R}$$- and $$\forall \exists \mathbb {R}$$-completeness of variants of Area Universality. To do so, we also introduce first tools to study $$\forall \exists \mathbb {R}$$. 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.

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© Springer Nature Switzerland AG 2018

## Authors and Affiliations

• Michael Gene Dobbins
• 1
• Linda Kleist
• 2
Email author
• Tillmann Miltzow
• 3
• Paweł Rzążewski
• 4
1. 1.Binghamton UniversityBinghamtonUSA
2. 2.Technische Universität BerlinBerlinGermany
3. 3.Université libre de BruxellesBrusselsBelgium
4. 4.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland

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