Optimal Curve Straightening is $\exists\mathbb{R}$-Complete

Erickson, Jeff

doi:10.48550/arxiv.1908.09400

Cited by 4 publications

(3 citation statements)

References 51 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Early examples are related to the recognition of geometric structures: points in the plane [43,57], geometric linkages [51], segment graphs [37,41], unit disk graphs [42,35], ray intersection graphs [21], and point visibility graphs [22]. In general, the complexity class is more established in the graph drawing community [38,25,49,28]. Yet, it is also relevant for studying polytopes [48,26].…”

Section: Scopementioning

confidence: 99%

On Classifying Continuous Constraint Satisfaction Problems

Miltzow,

Schmiermann

2024

TheoretiCS

View full text Add to dashboard Cite

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 $\exists x_1, \ldots, x_n \in \mathbb{R} : \Phi(x_1, \ldots, x_n)$, 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 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.

show abstract

Section: Scopementioning

confidence: 99%

On Classifying Continuous Constraint Satisfaction Problems

Miltzow,

Schmiermann

2024

TheoretiCS

View full text Add to dashboard Cite

show abstract

“…Important ∃R-completeness results include the realizability of abstract order types [58,78] and geometric linkages [70], as well as the recognition of geometric segment [52,55], unit-disk [50,56], and ray intersection graphs [17]. More results appeared in the graph drawing community [27,30,54,71], regarding polytopes [26,67], the study of Nashequilibria [10,12,13,33,72], matrix factorization [20,73,76,77], or continuous constraint satisfaction problems [57]. In computational geometry, we would like to mention the art gallery problem [2] and covering polygons with convex polygons [1].…”

Section: Existential Theory Of the Realsmentioning

confidence: 99%

Training Fully Connected Neural Networks is $\exists\mathbb{R}$-Complete

Bertschinger¹,

Hertrich²,

Jungeblut³

et al. 2022

Preprint

View full text Add to dashboard Cite

We consider the algorithmic problem of finding the optimal weights and biases for a twolayer fully connected neural network to fit a given set of data points. This problem is known as empirical risk minimization in the machine learning community. We show that the problem is ∃R-complete. This complexity class can be defined as the set of algorithmic problems that are polynomial-time equivalent to finding real roots of a polynomial with integer coefficients. Our results hold even if the following restrictions are all added simultaneously.• There are exactly two output neurons.• There are exactly two input neurons.• The data has only 13 different labels.• The number of hidden neurons is a constant fraction of the number of data points.• The target training error is zero.• The ReLU activation function is used.

show abstract

“…Early examples are related to recognition of geometric structures: points in the plane [36,49], geometric linkages [42,1], segment graphs [30,34], unit disk graphs [35,29], ray intersection graphs [17], and point visibility graphs [18]. In general, the complexity class is more established in the graph drawing community [31,20,43,23]. Yet, it is also relevant for studying polytopes [40,21].…”

Section: Existential Theory Of the Realsmentioning

confidence: 99%

On Classifying Continuous Constraint Satisfaction Problems

Miltzow¹,

Schmiermann²

2021

Preprint

View full text Add to dashboard Cite

A continuous constraint satisfaction problem (CCSP) is a constraint satisfaction problem (CSP) with an interval domain U ⊂ R. We engage in a systematic study to classify CCSPs that are complete of the Existential Theory of the Reals, i.e., ∃R-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 ∃x1, . . . , xn ∈ R : Φ(x1, . . . , xn), where Φ is a well-formed quantifier-free formula consisting of the symbols {0, 1, x1, . . . , xn, +, •, ≥, >, ∧, ∨, ¬}, the goal is to check whether this sentence is true. Now the class ∃R is the family of all problems that admit a polynomial-time many-one reduction to ETR. It is known that NP ⊆ ∃R ⊆ 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 • y = z), squaring constraints (x 2 = y), or inversion constraints (x • y = 1) are sufficient to establish ∃R-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 ∃R-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) ≥ 0 and g(x, y) ≥ 0) imply ∃R-completeness on the class of such CCSPs.Here, we call a function f : U 2 → R well-behaved if it is a C 2 -function, f (0, 0) = 0, all its partial derivatives fx, fy, f are rational in (0, 0), fx(0, 0) = 0 or fy(0, 0) = 0, and it can be computed on a real RAM. Furthermore we call f curved if the curvature of the curve given by f (x, y) = 0 is nonzero, at the origin. In this case we call f either convexly curved if the curvature is negative, or concavely curved if it is positive.We apply our findings to geometric packing and answer an open question by Abrahamsen et al. [5, FOCS 2020]. Namely, we establish ∃R-completeness of packing convex pieces into a square container under rotations and translations.

show abstract

Optimal Curve Straightening is $\exists\mathbb{R}$-Complete

Cited by 4 publications

References 51 publications

On Classifying Continuous Constraint Satisfaction Problems

On Classifying Continuous Constraint Satisfaction Problems

Training Fully Connected Neural Networks is $\exists\mathbb{R}$-Complete

On Classifying Continuous Constraint Satisfaction Problems

Contact Info

Product

Resources

About