# The Constraint Satisfaction Problem: Complexity and Approximability

@inproceedings{Fraser2017TheCS, title={The Constraint Satisfaction Problem: Complexity and Approximability}, author={A. S. Fraser and Andrei A. Krokhin}, booktitle={The Constraint Satisfaction Problem}, year={2017} }

Constraint satisfaction has always played a central role in computational complexity theory; appropriate versions of CSPs are classical complete problems for most standard complexity classes. CSPs constitute a very rich and yet sufficiently manageable class of problems to give a good perspective on general computational phenomena. For instance, they help to understand which mathematical properties make a computational problem tractable (in a wide sense, e.g., polynomial-time solvable, non…

## 32 Citations

Algebraic approach to promise constraint satisfaction

- Computer ScienceSTOC
- 2019

A new class of problems that can be viewed as algebraic versions of the (Gap) Label Cover problem are introduced, and it is shown that every PCSP with a fixed constraint language is equivalent to a problem of this form.

A Proof of the CSP Dichotomy Conjecture

- Mathematics, Computer ScienceJ. ACM
- 2020

This article presents an algorithm 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.

Fine-Grained Time Complexity of Constraint Satisfaction Problems

- Computer Science, MathematicsACM Trans. Comput. Theory
- 2021

Under the exponential-time hypothesis (ETH), the existence of subexponential algorithms for finite-domain NP-complete CSPΓ problems is ruled out and a relation SD is identified such that the time complexity of the NP- complete CSP({SD}) is a lower bound for all NP- Complete CSPs of this kind.

Algebraic Theory of Promise Constraint Satisfaction Problems, First Steps

- Mathematics, Computer ScienceFCT
- 2019

This paper explains an extension of this theory to a much broader class of computational problems, the promise CSPs, which includes relaxed versions of C SPs such as the problem of finding a 137-coloring of a 3-colorable graph.

The Complexity of the Distributed Constraint Satisfaction Problem

- Computer Science, MathematicsSTACS
- 2021

The results endorse the well-known fact from classical CSPs that the complexity of fixed-template computational problems depends on the template’s invariance under certain operations, and show that DCSP(Γ) is polynomial-time tractable if and only if Γ is invariant under symmetric polymorphisms of all arities.

The (Coarse) Fine-Grained Structure of NP-Hard SAT and CSP Problems

- Mathematics, Computer ScienceACM Trans. Comput. Theory
- 2022

This work finds the first example of an NP-complete SAT problem with a non-trivial algorithm which also admits aNon-Trivial lower bound under SETH, and suggests a dichotomy conjecture with a close connection to the CSP dichotomy theorem.

On the complexity of CSP-based ideal membership problems

- Mathematics, Computer ScienceSTOC
- 2022

A variation of the IMP is introduced and a unified framework, different from the celebrated Buchberger’s algorithm, is proposed to construct a bounded degree Gröbner Basis for many combinatorial problems.

A dichotomy theorem for nonuniform CSPs simplified

- MathematicsArXiv
- 2020

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

Minimal Taylor Algebras as a Common Framework for the Three Algebraic Approaches to the CSP

- Computer Science2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2021

The theory initiated in this paper will eventually result in a simple and more natural proof of the Dichotomy Theorem that employs a simpler and more efficient algorithm, and will help in attacking complexity questions in other CSP-related problems.

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.

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