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Approximating the Existential Theory of the Reals

  • Argyrios Deligkas
  • John Fearnley
  • Themistoklis MelissourgosEmail author
  • Paul G. Spirakis
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11316)

Abstract

The existential theory of the reals (ETR) consists of existentially quantified boolean formulas over equalities and inequalities of real-valued polynomials. We propose the approximate existential theory of the reals (\(\epsilon \)-ETR), in which the constraints only need to be satisfied approximately. We first show that unconstrained \(\epsilon \)-ETR = ETR, and then study the \(\epsilon \)-ETR problem when the solution is constrained to lie in a given convex set. Our main theorem is a sampling theorem, similar to those that have been proved for approximate equilibria in normal form games. It states that if an ETR problem has an exact solution, then it has a k-uniform approximate solution, where k depends on various properties of the formula. A consequence of our theorem is that we obtain a quasi-polynomial time approximation scheme (QPTAS) for a fragment of constrained \(\epsilon \)-ETR. We use our theorem to create several new PTAS and QPTAS algorithms for problems from a variety of fields.

Notes

Acknowledgements

P. Spirakis wishes to dedicate this paper to the memory of his late father in law Mathematician and Professor Dimitrios Chrysofakis, who was among the first in Greece to work on tensor analysis.

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Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Argyrios Deligkas
    • 1
    • 2
  • John Fearnley
    • 1
  • Themistoklis Melissourgos
    • 1
    Email author
  • Paul G. Spirakis
    • 1
    • 3
  1. 1.Department of Computer ScienceUniversity of LiverpoolLiverpoolUK
  2. 2.Leverhulme Research Centre for Functional Materials DesignLiverpoolUK
  3. 3.Computer Engineering and Informatics DepartmentUniversity of PatrasPatrasGreece

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