Smoothing the Gap Between NP and ER

Erickson, Jeff; Hoog, Ivor van der; Miltzow, Tillmann

doi:10.1137/20m1385287

Cited by 6 publications

(8 citation statements)

References 48 publications

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“…Proof. By a result from Erickson, van der Hoog and Miltzow we can prove ∃R-membership by describing a polynomial-time verification algorithm for a real RAM machine 3 [10]. Given a graph G = (V, E) and for each vertex v ∈ V a point (v x , v y , v z ) in the hyperboloid model of the hyperbolic plane representing the center of an equal-radius disk.…”

Section: Discussionmentioning

confidence: 99%

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Recognizing Unit Disk Graphs in Hyperbolic Geometry is $\exists\mathbb{R}$-Complete

Bieker¹,

Bläsius²,

Dohse³

et al. 2023

Preprint

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A graph G is a (Euclidean) unit disk graph if it is the intersection graph of unit disks in the Euclidean plane R 2 . Recognizing them is known to be ∃R-complete, i.e., as hard as solving a system of polynomial inequalities. In this note we describe a simple framework to translate ∃R-hardness reductions from the Euclidean plane R 2 to the hyperbolic plane H 2 . We apply our framework to prove that the recognition of unit disk graphs in the hyperbolic plane is also ∃R-complete.

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Section: Discussionmentioning

confidence: 99%

“…However, arbitrary analytic functions (like arcosh) are not supported. See[10] for a formal definition.…”

mentioning

confidence: 99%

Recognizing Unit Disk Graphs in Hyperbolic Geometry is $\exists\mathbb{R}$-Complete

Bieker¹,

Bläsius²,

Dohse³

et al. 2023

Preprint

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“…We are mainly interested in CCSPs where the constraints are semi-algebraic over the integers (see Section 1.2 for a formal definition). There are some constraints that are not semialgebraic, in other words, not computable on the real RAM [29]. For example, constraints involving sin, cos, exp, log or testing if a number is an integer.…”

Section: E F I N I T I O N 1 4 (Constraint Satisfaction Problem)mentioning

confidence: 99%

“…Given a point 𝑥 ∈ R 𝑛 and a semi-algebraic set 𝑆 ⊂ R 𝑛 , we can decide on the real RAM if 𝑥 ∈ 𝑆. (We refer the reader to the work by Erickson, Hoog, and Miltzow for a detailed definition of the real RAM and decidability [29].) This is easy to see as we only need to evaluate the defining formula of 𝑆. Interestingly the reverse direction also holds.…”

Section: Existential Theory Of the Realsmentioning

confidence: 99%

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On Classifying Continuous Constraint Satisfaction Problems

Miltzow,

Schmiermann

2024

TheoretiCS

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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.

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ReLU neural networks of polynomial size for exact maximum flow computation

Hertrich,

Sering

2024

Math. Program.

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Smoothing the Gap Between NP and ER

Cited by 6 publications

References 48 publications

Recognizing Unit Disk Graphs in Hyperbolic Geometry is $\exists\mathbb{R}$-Complete

Recognizing Unit Disk Graphs in Hyperbolic Geometry is $\exists\mathbb{R}$-Complete

On Classifying Continuous Constraint Satisfaction Problems

ReLU neural networks of polynomial size for exact maximum flow computation

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