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.

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逼近实数的存在论

实数存在论 (ETR) 由 $\mathbb{R}$ 中变量多项式函数的等式和不等式的存在量化布尔公式组成。在本文中，我们提出并研究了实数的近似存在论（$\epsilon$-ETR），其中约束只需要近似地满足。我们首先证明当变量的定义域为 $\mathbb{R}$ 时，$\epsilon$-ETR = ETR 在多项式时间约简下，然后研究当变量被约束为时的约束 $\epsilon$-ETR 问题位于给定的有界凸集。我们的主要定理是一个抽样定理，类似于在正规形式博弈中为近似均衡证明的那些定理。它以网格状方式离散域，其密度取决于公式的各种属性。我们的定理的一个结果是，我们为受约束的 $\epsilon$-ETR 的片段获得了一个准多项式时间近似方案 (QPTAS)。我们使用我们的定理为来自各个领域的问题创建了几个新的 PTAS 和 QPTAS 算法。