$\forall \exists \mathbb{R}$-completeness and area-universality

In the study of geometric problems, the complexity class $\exists \mathbb{R}$ 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, $\exists \mathbb{R}$ deals with existentially quantified real variables. In analogy to $\Pi_2^p$ and $\Sigma_2^p$ in the famous polynomial hierarchy, we introduce and motivate 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 $\forall\exists \mathbb{R}$, such as restricted variants of UETR, which are $\forall\exists \mathbb{R}$-complete. 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.

[1]  Therese C. Biedl,et al.  Drawing planar 3-trees with given face areas , 2013, Comput. Geom..

[2]  Linda Kleist Drawing Planar Graphs with Prescribed Face Areas , 2016, WG.

[3]  Tillmann Miltzow Augmenting a Geometric Matching is NP-complete , 2012, ArXiv.

[4]  Maria Belk,et al.  Realizability of Graphs in Three Dimensions , 2007, Discret. Comput. Geom..

[5]  Günter Rote,et al.  Blowing Up Polygonal Linkages , 2003 .

[6]  James H. Davenport,et al.  Real Quantifier Elimination is Doubly Exponential , 1988, J. Symb. Comput..

[7]  Gunter M. Ziegler,et al.  Realization spaces of 4-polytopes are universal , 1995 .

[8]  J. Kratochvil,et al.  Intersection Graphs of Segments , 1994, J. Comb. Theory, Ser. B.

[9]  Paul W. Goldberg,et al.  The Complexity of Computing a Nash Equilibrium , 2009, SIAM J. Comput..

[10]  Lenore Blum,et al.  Complexity and Real Computation , 1997, Springer New York.

[11]  Jrgen Richter-Gebert,et al.  Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry , 2011 .

[12]  Carsten Thomassen Plane Cubic Graphs with Prescribed Face Areas , 1992, Comb. Probab. Comput..

[13]  Marcus Schaefer,et al.  Realizability of Graphs and Linkages , 2013 .

[14]  Robert Connelly,et al.  Realizability of Graphs , 2007, Discret. Comput. Geom..

[15]  Paul D. Seymour,et al.  Graph Minors. XX. Wagner's conjecture , 2004, J. Comb. Theory B.

[16]  Bruce Bueno de Mesquita,et al.  An Introduction to Game Theory , 2014 .

[17]  Tillmann Miltzow,et al.  The Art Gallery Problem is $\exists \mathbb{R}$-complete , 2017 .

[18]  N. Mnev The universality theorems on the classification problem of configuration varieties and convex polytopes varieties , 1988 .

[19]  S. Basu,et al.  Algorithms in real algebraic geometry , 2003 .

[20]  Riste Škrekovski,et al.  Coloring face hypergraphs on surfaces , 2005, Eur. J. Comb..

[21]  Marcus Schaefer,et al.  Fixed Points, Nash Equilibria, and the Existential Theory of the Reals , 2017, Theory of Computing Systems.