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Completeness for the Complexity Class

Michael Gene Dobbins1, Linda Kleist2, Tillmann Miltzow3

  • 1Binghamton University, Binghamton, USA.

Discrete & Computational Geometry
|June 9, 2023
PubMed
Summary
This summary is machine-generated.

This study explores the complexity class ∃R, the real analog of NP, and its role in geometric problems. Researchers investigate the Area Universality problem, proposing it as ∀∃R-complete and proving related problems are ∃R- and ∀∃R-complete.

Keywords:
Area-universalityComplexity classExistential theory of the realsFace areaPlanar graphUniversal existential theory of the reals

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Area of Science:

  • Computational geometry
  • Real algebraic complexity theory

Background:

  • The complexity class ∃R, analogous to NP but with real variables, is central to geometric problems.
  • The polynomial hierarchy, including Π2p and Σ2p, provides a framework for studying complex computational problems.

Purpose of the Study:

  • To investigate the complexity of the Area Universality problem in geometric graph drawing.
  • To explore and establish the ∀∃R-completeness of the Area Universality problem and related geometric challenges.

Main Methods:

  • Analysis of the complexity class ∃R and its extensions, ∀∃R and ∃∀R, using real variables.
  • Development of novel techniques for proving ∀∃R-hardness and membership.
  • Reduction of geometric problems to establish complexity class completeness.

Main Results:

  • The Area Universality problem is conjectured to be ∀∃R-complete.
  • Two variants of the Area Universality problem are proven to be ∃R-complete and ∀∃R-complete.
  • New tools for analyzing ∀∃R-hardness and membership are introduced.

Conclusions:

  • The study provides significant insights into the computational complexity of geometric problems involving real numbers.
  • Geometric problems related to imprecision, robustness, and extendability are identified as potential ∀∃R-complete problems.