∀ ∃ R -completeness and area-universality
Michael Gene Dobbins, Linda Kleist, Tillmann Miltzow, Paweł Rzążewski · Dépôt institutionnel de l'Université libre de Bruxelles (Université Libre de Bruxelles) · 2018
In the study of geometric problems, the complexity class ∃ R plays a crucial role since it exhibits a deep connection between purely geometric problems and real algebra. Sometimes ∃ 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, ∃ R deals with existentially quantified real variables. In analogy to Π2p and Σ2p in the famous polynomial hierarchy, we study the complexity classes ∀ ∃ R and ∃ ∀ 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 ∀ ∃ R -complete and support this conjecture by a series of partial results, where we prove ∃ R - and ∀ ∃ R -completeness of variants of Area Universality. To do so, we also introduce first tools to study ∀ ∃ R. Finally, we present geometric problems as candidates for ∀ ∃ R -complete problems. These problems have connections to the concepts of imprecision, robustness, and extendability.