On Classifying Continuous Constraint Satisfaction Problems
Tillmann Miltzow, Reinier F. Schmiermann
TL;DR
This work classifies continuous constraint satisfaction problems (CCSPs) in the real domain through the lens of ER-completeness, introducing CE (curved equality) and CCI (convex-concave inequality) frameworks. The authors develop a robust reduction toolkit, including ETR-CONJ, ETR-COMPACT, ETR-AMI, ETR-SMALL, and ETR-SQUARE variants, to prove ER-hardness from well-behaved, nonlinear constraints paired with addition. They show that any CCSP with an addition constraint plus a single well-behaved curved equality constraint is ER-complete, and that well-behaved convexly curved or concavely curved inequalities also yield ER-completeness, enabling broad applicability to geometric packing and related problems. The paper further provides explicit construction techniques (approximation, explicit/implicit representations) and discusses alternative constraint descriptions, promise problems, and potential generalizations, highlighting the deep connections between CCSPs and the existential theory of the reals. Overall, the results offer a unified framework for proving ER-hardness across continuous constraint families and illuminate why certain geometric and algebraic problems resist efficient discretization.
Abstract
A continuous constraint satisfaction problem (CCSP) is a constraint satisfaction problem (CSP) with an interval 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 $\exists x_1, \ldots, x_n \in \mathbb{R} : Φ(x_1, \ldots, x_n)$, where $Φ$ 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 many-one 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 conditions. 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 conditions) 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.