Completeness for the Complexity Class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \exists \mathbb {R}$$\end{document}∀∃R and Area-Universality

• Published in: Discrete & computational geometry Vol. 70; no. 1; pp. 154 - 188
• Main Authors: Dobbins, Michael Gene, Kleist, Linda, Miltzow, Tillmann, Rzążewski, Paweł
• Format: Journal Article
• Language: English
• Published: New York Springer US 18.05.2022
• ISSN: 0179-5376; 1432-0444
• DOI: 10.1007/s00454-022-00381-0

Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists \mathbb {R}$$\end{document} ∃ R plays a crucial role in the study of geometric problems. Sometimes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists \mathbb {R}$$\end{document} ∃ 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, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists \mathbb {R}$$\end{document} ∃ R deals with existentially quantified real variables. In analogy to \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Pi _2^p$$\end{document} Π 2 p and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Sigma _2^p$$\end{document} Σ 2 p in the famous polynomial hierarchy, we study the complexity classes \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \exists \mathbb {R}$$\end{document} ∀ ∃ R and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \exists \forall \mathbb {R}$$\end{document} ∃ ∀ 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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \exists \mathbb {R}$$\end{document} ∀ ∃ R -complete and support this conjecture by proving \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists \mathbb {R}$$\end{document} ∃ R - and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \exists \mathbb {R}$$\end{document} ∀ ∃ R -completeness of two variants of Area Universality . To this end, we introduce tools to prove \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \exists \mathbb {R}$$\end{document} ∀ ∃ R -hardness and membership. Finally, we present geometric problems as candidates for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\forall \exists \mathbb {R}$$\end{document} ∀ ∃ R -complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability.