On Classifying Continuous Constraint Satisfaction problems

A continuous constraint satisfaction problem (CCSP) is a constraint satisfaction problem (CSP) with a domain $U \subset \mathbb{R}$. We engage in a systematic study to classify CCSPs that are complete of the Existential Theory of the Reals, i.e., ER-complete. To define this class, we first consider the problem ETR, which also stands for Existential Theory of the Reals. In an instance of this problem we are given some sentence of the form $\exists x_1, \ldots, x_n \in \mathbb{R} : \Phi(x_1, \ldots, x_n)$, where $\Phi$ is a well-formed quantifier-free formula consisting of the symbols $\{0, 1, +, \cdot, \geq, >, \wedge, \vee, \neg\}$, the goal is to check whether this sentence is true. Now the class ER is the family of all problems that admit a polynomial-time reduction to ETR. It is known that NP $\subseteq$ ER $\subseteq$ PSPACE. We restrict our attention on CCSPs with addition constraints ($x + y = z$) and some other mild technical condition. Previously, it was shown that multiplication constraints ($x \cdot y = z$), squaring constraints ($x^2 = y$), or inversion constraints ($x\cdot y = 1$) are sufficient to establish ER-completeness. We extend this in the strongest possible sense for equality constraints as follows. We show that CCSPs (with addition constraints and some other mild technical condition) that have any one well-behaved curved equality constraint ($f(x,y) = 0$) are ER-complete. We further extend our results to inequality constraints. We show that any well-behaved convexly curved and any well-behaved concavely curved inequality constraint ($f(x,y) \geq 0$ and $g(x,y) \geq 0$) imply ER-completeness on the class of such CCSPs. We apply our findings to geometric packing and answer an open question by Abrahamsen et al. [FOCS 2020]. Namely, we establish ER-completeness of packing convex pieces into a square container under rotations and translations.