\(\forall \exists \mathbb {R}\)-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)


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.


  1. 1.
    Abrahamsen, M., Adamaszek, A., Miltzow, T.: The art gallery problem is \(\exists \mathbb{R}\)-complete. In: Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing. ACM (2018)Google Scholar
  2. 2.
    Basu, S., Pollack, R., Roy, M.-F.: Algorithms in Real Algebraic Geometry. Springer, Heidelberg (2006). Scholar
  3. 3.
    Biedl, T.C., Velázquez, L.E.R.: Drawing planar 3-trees with given face areas. Comput. Geom. 46(3), 276–285 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, New York (2012). Scholar
  5. 5.
    Davenport, J.H., Heintz, J.: Real quantifier elimination is doubly exponential. J. Symb. Comput. 5(1), 29–35 (1988)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Dobbins, M.G., Kleist, L., Miltzow, T., Rzążewski, P.: \(\forall \)\(\exists \)R-completeness and area-universality. CoRR, abs/1712.05142 (2017)Google Scholar
  7. 7.
    Dvořák, Z., Král’, D., Škrekovski, R.: Coloring face hypergraphs on surfaces. Eur. J. Comb. 26(1), 95–110 (2005)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Evans, W.S., et al.: Table cartogram. Comput. Geom. 68, 174–185 (2018)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kleist, L.: Drawing planar graphs with prescribed face areas. J. Comput. Geom. 9(1), 290–311 (2018)zbMATHGoogle Scholar
  10. 10.
    Levi, F.: Die Teilung der projektiven Ebene durch Gerade oder Pseudogerade. Ber. Math.-Phys. Kl. Sächs. Akad. Wiss 78, 256–267 (1926)Google Scholar
  11. 11.
    Lubiw, A., Miltzow, T., Mondal, D.: The complexity of drawing a graph in a polygonal region. CoRR, abs/1802.06699 (2018). Accepted at Graph Drawing 2018 (GD 2018)Google Scholar
  12. 12.
    Matoušek, J.: Intersection graphs of segments and \(\exists \mathbb{R}\). CoRR, abs/1406.2636 (2014)Google Scholar
  13. 13.
    Mnev, N.E.: The universality theorems on the classification problem of configuration varieties and convex polytopes varieties. In: Viro, O.Y., Vershik, A.M. (eds.) Topology and Geometry—Rohlin Seminar. LNM, vol. 1346, pp. 527–543. Springer, Heidelberg (1988). Scholar
  14. 14.
    Richter-Gebert, J.: Perspectives on Projective Geometry: A Guided Tour Through Real and Complex Geometry. Springer, Heidelberg (2011). Scholar
  15. 15.
    Ringel, G.: Equiareal graphs. In: Contemporary Methods in Graph Theory, pp. 503–505 (1990)Google Scholar
  16. 16.
    Schaefer, M., Štefankovič, D.: Fixed points, Nash equilibria, and the existential theory of the reals. Theory Comput. Syst. 60(2), 172–193 (2017)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Thomassen, C.: Plane cubic graphs with prescribed face areas. Comb. Probab. Comput. 1, 371–381 (1992)MathSciNetCrossRefGoogle Scholar

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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