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

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.

A video presenting this paper is available at https://youtu.be/OQkACiNS66o. Proofs omitted due to space constraints can be found in the full version of the manuscript [6]

T. Miltzow—Partially supported by the ERC grant PARAMTIGHT: “Parameterized complexity and the search for tight complexity results”, no. 280152.