# On Classifying Continuous Constraint Satisfaction problems

@article{Miltzow2022OnCC, title={On Classifying Continuous Constraint Satisfaction problems}, author={Tillmann Miltzow and Reinier F. Schmiermann}, journal={2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)}, year={2022}, pages={781-791} }

A continuous constraint satisfaction problem (CCSP) is a constraint satisfaction problem (CSP) with an interval domain <tex>$U\subset \mathbb{R}$</tex>. We engage in a systematic study to classify CCSPs that are complete of the Existential Theory of the Reals, i.e., <tex>$\exists \mathbb{R}$</tex> -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 <tex…

## 8 Citations

### Smoothing the gap between NP and ER

- Computer Science, Mathematics2020 IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS)
- 2020

A real RAM analogue to the Cook-Levin theorem is proved which shows that ER membership is equivalent to having a verification algorithm that runs in polynomial-time on a real RAM, which gives an easy proof of ER-membership.

### Geometric Embeddability of Complexes is $\exists \mathbb R$-complete

- Mathematics
- 2021

We show that the decision problem of determining whether a given (abstract simplicial) k-complex has a geometric embedding in R is complete for the Existential Theory of the Reals for all d ≥ 3 and k…

### Training Neural Networks is $\exists\mathbb R$-complete

- Computer Science
- 2021

The result of this paper offers an explanation why techniques commonly used to solve big instances of NP-complete problems seem not to be of use for this task, by showing that ∃R-complete.

### Training Neural Networks is ∃R-complete

- Computer Science
- 2021

The result of this paper offers an explanation why techniques commonly used to solve big instances of NP-complete problems seem not to be of use for this task, by showing that ∃R-complete.

### Topological Art in Simple Galleries

- MathematicsSOSA
- 2022

It is shown that for every semi-algebraic set S there is a polygon P such that V (P ) is homotopy equivalent to S, and that for various concrete topological spaces T, instances I of the art gallery problem such as V (I) is homeomorphic to T.

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

- Computer Science
- 2022

The algorithmic problem of finding the optimal weights and biases for a two-layer fully connected neural network to a given set of data points is considered and it is shown that even very simple networks are difficult to train.

### Training Fully Connected Neural Networks is ∃R-Complete

- Computer ScienceArXiv
- 2022

The algorithmic problem of finding the optimal weights and biases for a two-layer fully connected neural network to a given set of data points is considered and it is shown that even very simple networks are difficult to train.

### The Complexity of the Hausdorff Distance

- MathematicsSoCG
- 2022

It is shown that the decision problem of whether the Hausdorff distance of two semi-algebraic sets is bounded by a given threshold is complete for the complexity class ∀∃<R.hard, which implies that the problem is NP-, co-NP-, ∃Rand ∀R-hard.

## References

SHOWING 1-10 OF 61 REFERENCES

### Covering Polygons is Even Harder

- Mathematics, Computer Science2021 IEEE 62nd Annual Symposium on Foundations of Computer Science (FOCS)
- 2022

It is proved that assuming the widespread belief that NP-hard MCC is not in N P, and the problem is thus $\exists \mathbb{R}$-complete, that many natural approaches to finding small covers are bound to give suboptimal solutions in some cases.

### The complexity of temporal constraint satisfaction problems

- Computer ScienceJACM
- 2010

This work presents a complete complexity classification of the constraint satisfaction problem (CSP) for temporal constraint languages: if the constraint language is contained in one out of nine temporal constraint language, then the CSP can be solved in polynomial time; otherwise, the C SP is NP-complete.

### A Dichotomy Theorem for Nonuniform CSPs

- Mathematics2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
- 2017

The Dichotomy Conjecture for the non-uniform CSP states that for every constraint language \Gm the problem CSP is either solvable in polynomial time or is NP-complete.

### A dichotomy theorem for constraint satisfaction problems on a 3-element set

- Computer Science, MathematicsJACM
- 2006

Every subproblem of the CSP is either tractable or NP-complete, and the criterion separating them is that conjectured in Bulatov et al.

### A Proof of CSP Dichotomy Conjecture

- Mathematics, Computer Science2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)
- 2017

An algorithm is presented that solves Constraint Satisfaction Problem in polynomial time for constraint languages having a weak near unanimity polymorphism, which proves the remaining part of the conjecture.

### Approximating the Existential Theory of the Reals

- Mathematics, Computer ScienceWINE
- 2018

The main theorem is a sampling theorem, similar to those that have been proved for approximate equilibria in normal form games, that states that if an ETR problem has an exact solution, then it has a k-uniform approximate solution, where k depends on various properties of the formula.

### Constraint Satisfaction Problems over Numeric Domains

- Computer ScienceThe Constraint Satisfaction Problem
- 2017

A survey of complexity results for constraint satisfaction problems (CSPs) over the integers, the rationals, the reals, and the complex numbers is presented, to identify those CSPs that can be solved in polynomial time, and to distinguish them from C SPs that are NP-hard.

### Realizability of Graphs and Linkages

- Mathematics
- 2013

We show that deciding whether a graph with given edge lengths can be realized by a straight-line drawing has the same complexity as deciding the truth of sentences in the existential theory of the…

### The complexity of satisfiability problems

- MathematicsSTOC
- 1978

An infinite class of satisfiability problems is considered which contains these two particular problems as special cases, and it is shown that every member of this class is either polynomial-time decidable or NP-complete.