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- W2894903419 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." @default.
- W2894903419 created "2018-10-12" @default.
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- W2894903419 date "2018-01-01" @default.
- W2894903419 modified "2023-09-30" @default.
- W2894903419 title "Approximating the Existential Theory of the Reals" @default.
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- W2894903419 doi "https://doi.org/10.1007/978-3-030-04612-5_9" @default.
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