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 ($ε$-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 $ε$-ETR = ETR under polynomial time reductions, and then study the constrained $ε$-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 $ε$-ETR. We use our theorem to create several new PTAS and QPTAS algorithms for problems from a variety of fields.

In the proceedings of the 14th Conference on Web and Internet Economics (WINE 2018)