# A Proof of CSP Dichotomy Conjecture

@article{Zhuk2017APO, title={A Proof of CSP Dichotomy Conjecture}, author={Dmitriy Zhuk}, journal={2017 IEEE 58th Annual Symposium on Foundations of Computer Science (FOCS)}, year={2017}, pages={331-342} }

Many natural combinatorial problems can be expressed as constraint satisfaction problems. This class of problems is known to be NP-complete in general, but certain restrictions on the form of the constraints can ensure tractability. The standard way to parametrize interesting subclasses of the constraint satisfaction problem is via finite constraint languages. The main problem is to classify those subclasses that are solvable in polynomial time and those that are NP-complete. It was conjecturedâ€¦Â

## 297 Citations

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 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.

Descriptive complexity of constraint problems

- Computer Science
- 2018

A dichotomy on the number of levels in the Lasserre hierarchy necessary to obtain an exact solution is proved, which proves that a dichotomy exists also in the general case of CSPs, and several results on the definability of VCSPs are obtained.

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.

Topology is relevant (in a dichotomy conjecture for infinite-domain constraint satisfaction problems)

- Mathematics2019 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2019

It is shown that local satisfaction and global satisfaction of nontrivial height 1 identities differ for $\omega$ -categorical structures with less than double exponential orbit growth, thereby resolving one of the main open problems in the algebraic theory of such structures.

Universal Algebraic Methods for Constraint Satisfaction Problems

- Mathematics, Computer ScienceLog. Methods Comput. Sci.
- 2022

The utility of new techniques that help attack the class of finite algebras known as "commutative idempotent binars" (CIBs) are demonstrated by using them to prove that every CIB of cardinality at most 4 yields a tractable CSP.

Topology is relevant (in the infinite-domain dichotomy conjecture for constraint satisfaction problems)

- MathematicsArXiv
- 2019

It is shown that local satisfaction and global satisfaction of non-trivial height 1 identities differ for $\omega$-categorical structures with less than double exponential orbit growth, thereby resolving one of the main open problems in the algebraic theory of such structures.

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.

Complexity of Infinite-Domain Constraint Satisfaction

- Computer Science
- 2021

This monograph presents a self-contained introduction to the universal-algebraic approach to complexity classification, treating both finite and infinite-domain CSPs.

Proof complexity of CSP on algebras with linear congruence

- Mathematics, Computer Science
- 2022

Zhukâ€™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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