Approximating the Existential Theory of the Reals



Deligkas, Argyrios, Fearnley, John, Melissourgos, Themistoklis ORCID: 0000-0002-9867-6257 and Spirakis, Paul G
(2018) Approximating the Existential Theory of the Reals. .

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Abstract

The Existential Theory of the Reals (ETR) consists of existentially quantified Boolean formulas over equalities and inequalities of polynomial functions of variables in $\mathbb{R}$. In this paper we propose and study the approximate existential theory of the reals ($\epsilon$-ETR), in which the constraints only need to be satisfied approximately. We first show that when the domain of the variables is $\mathbb{R}$ then $\epsilon$-ETR = ETR under polynomial time reductions, and then study the constrained $\epsilon$-ETR problem when the variables are constrained to lie in a given bounded convex set. Our main theorem is a sampling theorem, similar to those that have been proved for approximate equilibria in normal form games. It discretizes the domain in a grid-like manner whose density 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.

Item Type: Conference or Workshop Item (Unspecified)
Additional Information: In the proceedings of the 14th Conference on Web and Internet Economics (WINE 2018)
Uncontrolled Keywords: cs.CC, cs.CC, cs.CG, cs.GT, math.GN
Depositing User: Symplectic Admin
Date Deposited: 25 Sep 2018 15:32
Last Modified: 27 Apr 2022 14:10
Related URLs:
URI: https://livrepository.liverpool.ac.uk/id/eprint/3026725