# A Dichotomy Theorem for Nonuniform CSPs

@article{Bulatov2017ADT, title={A Dichotomy Theorem for Nonuniform CSPs}, author={Andrei A. Bulatov}, journal={2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)}, year={2017}, pages={319-330} }

In a non-uniform Constraint Satisfaction problem CSP(Γ), where G is a set of relations on a finite set A, the goal is to find an assignment of values to variables subject to constraints imposed on specified sets of variables using the relations from Γ. The Dichotomy Conjecture for the non-uniform CSP states that for every constraint language \Gm the problem CSP(Γ) is either solvable in polynomial time or is NP-complete. It was proposed by Feder and Vardi in their seminal…

## 299 Citations

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.

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.

Finitely Tractable Promise Constraint Satisfaction Problems

- Computer ScienceMFCS
- 2021

This work begins a systematic study of this phenomenon by giving a general necessary condition for finite tractability and characterizing finite tractable within a class of templates - the "basic" tractable cases in the dichotomy theorem for symmetric Boolean PCSPs allowing negations by Brakensiek and Guruswami.

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.

An Algebraic Approach to Valued Constraint Satisfaction

- Computer ScienceCSL
- 2017

A notion of polymorphism is introduced that captures the pp-definability in the style of Geiger’s result, and sufficient conditions for tractability of the classical CSP, related to the existence of certain polymorphisms, are shown to serve also for the valued case.

Fine-Grained Complexity of Constraint Satisfaction Problems through Partial Polymorphisms: A Survey

- Computer Science2019 IEEE 49th International Symposium on Multiple-Valued Logic (ISMVL)
- 2019

This contribution surveys an alternative approach based on partial polymorphisms, which is useful for studying the fine-grained complexity of NP-complete CSPs, and state some challenging open problems in the research field.

Time Complexity of Constraint Satisfaction via Universal Algebra

- Computer Science, MathematicsMFCS
- 2017

The worst-case time complexity of NP-complete CSPs, where one is allowed to arbitrarily restrict the values of individual variables, is studied, and it is proved that the complexity of CSP({SD}) is a lower bound for all C SPs of this particular kind.

Quantaloidal approach to constraint satisfaction

- MathematicsArXiv
- 2021

It is pointed out that fundamental concepts of the CSP can be formulated abstractly inside the 2-category PFinSet of finite sets and sets of functions between them, and their computational complexity can be classified by the associated notion of polymorphism.

A modifiction of the CSP algorithm for infinite languages

- Computer ScienceArXiv
- 2018

A modification of the algorithm that works in polynomial time even for infinite constraint languages is presented, which proves that if a constraint language has a weak near unanimity polymorphism then the corresponding constraint satisfaction problem is tractable, otherwise it is NP-complete.

Proof complexity of CSP on algebras with linear congruence

- Mathematics, Computer Science
- 2022

Zhuk’s algorithm for negative instances of the CSP problem can be augmented by extra information: it not only rejects X that cannot be homomorphically mapped into A, but produces a certificate a short extended Frege (EF) propositional proof that this rejection is correct.

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