Completeness for the Complexity Class $$\forall \exists \mathbb {R}$$ and Area-Universality

Autor: Michael Gene Dobbins, Linda Kleist, Tillmann Miltzow, Paweł Rzążewski
Rok vydání: 2022
Předmět:
Zdroj: Discrete & Computational Geometry.
ISSN: 1432-0444
0179-5376
DOI: 10.1007/s00454-022-00381-0
Popis: Exhibiting a deep connection between purely geometric problems and real algebra, the complexity class $\exists \mathbb{R}$ plays a crucial role in the study of geometric problems. Sometimes $\exists \mathbb{R}$ is referred to as the 'real analog' of NP. While NP is a class of computational problems that deals with existentially quantified boolean variables, $\exists \mathbb{R}$ deals with existentially quantified real variables. In analogy to $\Pi_2^p$ and $\Sigma_2^p$ in the famous polynomial hierarchy, we study the complexity classes $\forall \exists \mathbb{R}$ and $\exists \forall \mathbb{R}$ 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 $\forall \exists \mathbb{R}$-complete and support this conjecture by proving $\exists \mathbb{R}$- and $\forall \exists \mathbb{R}$-completeness of two variants of area-universality. To this end, we introduce tools to prove $\forall \exists \mathbb{R}$-hardness and membership. 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.
Comment: 36 pages, 17 figures
Databáze: OpenAIRE
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