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

  • Michael Gene Dobbins
  • Linda KleistEmail author
  • Tillmann Miltzow
  • Paweł Rzążewski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11159)

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.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Michael Gene Dobbins
    • 1
  • Linda Kleist
    • 2
    Email author
  • Tillmann Miltzow
    • 3
  • Paweł Rzążewski
    • 4
  1. 1.Binghamton UniversityBinghamtonUSA
  2. 2.Technische Universität BerlinBerlinGermany
  3. 3.Université libre de BruxellesBrusselsBelgium
  4. 4.Faculty of Mathematics and Information ScienceWarsaw University of TechnologyWarsawPoland

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