# A Proof of the CSP Dichotomy Conjecture

@article{Zhuk2020APO, title={A Proof of the CSP Dichotomy Conjecture}, author={Dmitriy Zhuk}, journal={Journal of the ACM (JACM)}, year={2020}, volume={67}, pages={1 - 78} }

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 parameterize 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…

## 14 Citations

The exponential-time hypothesis and the relative complexity of optimization and logical reasoning problems

- Computer ScienceTheor. Comput. Sci.
- 2021

Acyclic orders, partition schemes and CSPs: Unified hardness proofs and improved algorithms

- Computer ScienceArtif. Intell.
- 2021

When symmetries are not enough: a hierarchy of hard Constraint Satisfaction Problems

- MathematicsSIAM J. Comput.
- 2022

A model-theoretic construction -- a refinement of the Hrushosvki-encoding -- is applied to $\omega$-categorical structures, showing that the encoded structures retain desirable algebraic properties, but that the constraint satisfaction problems (CSPs) associated with these structures can be badly behaved in terms of computational complexity.

On approximability of satisfiable k-CSPs: I

- Mathematics, Computer ScienceSTOC
- 2022

It is shown that under certain conditions on P and a (1,s) integrality gap instance of the P-CSP problem, it can be translated into a dictatorship vs. quasirandomness test with perfect completeness and soundness s+ε, for every constant ε>0.

Galois Connections for Patterns: An Algebra of Labelled Graphs

- Computer Science, MathematicsGKR
- 2020

This work presents a Galois connection between lattices composed of sets of forbidden patterns and sets of generic instances, and investigates the power of forbidding augmented patterns and discusses their potential for describing new tractable classes.

The Complexity of Conjunctive Queries with Degree 2

- Computer Science, MathematicsPODS
- 2022

It is shown that that the tractability of conjunctive query answering can be characterised in terms of generalised hypertree width, and introduces hypergraph dilutions as an alternative method to primal graph minors for studying substructures of hypergraphs.

ℋ-Colouring Dichotomy in Proof Complexity

- Mathematics, Computer ScienceJ. Log. Comput.
- 2021

The main goal of this work is to start the development of some of the theories beyond the CSP dichotomy theorem in bounded arithmetic, and to formalize in such a theory the soundness of Zhuk’s algorithm, extending the upper bound proved here from undirected simple graphs to the general case of directed graphs in some logical calculi.

Sandwiches for promise constraint satisfaction

- Computer Science, Materials ScienceArXiv
- 2020

Promise Constraint Satisfaction Problems ( $$\mathrm{PCSP}$$ PCSP ) were proposed recently by Brakensiek and Guruswami as a framework to study approximations for Constraint Satisfaction Problems (…

On a stronger reconstruction notion for monoids and clones

- MathematicsForum Mathematicum
- 2021

Abstract Motivated by reconstruction results by Rubin, we introduce a new reconstruction notion for permutation groups, transformation monoids and clones, called automatic action compatibility, which…

## References

SHOWING 1-10 OF 80 REFERENCES

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.

Closure properties of constraints

- MathematicsJACM
- 1997

This paper investigates the subclasses that arise from restricting the possible constraint types, and shows that any set of constraints that does not give rise to an NP-complete class of problems must satisfy a certain type of algebraic closure condition.

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.

Classifying the Complexity of Constraints Using Finite Algebras

- MathematicsSIAM J. Comput.
- 2005

It is shown that any set of relations used to specify the allowed forms of constraints can be associated with a finite universal algebra and how the computational complexity of the corresponding constraint satisfaction problem is connected to the properties of this algebra is explored.

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.

The Constraint Satisfaction Problem: Complexity and Approximability

- Computer ScienceThe Constraint Satisfaction Problem
- 2017

This report documents the material presented during the course of the Dagstuhl Seminar 18231 “The Constraint Satisfaction Problem: Complexity and Approximability”, aimed at bringing together researchers using all the different techniques in the study of the CSP to share their insights obtained.

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.

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.

Complexity Classification in Infinite-Domain Constraint Satisfaction

- Computer ScienceArXiv
- 2012

This thesis studies CSPs where the variables can take values from an infinite domain, and studies the limits of complexity classification, and presents classes of computational problems provably do not exhibit a complexity dichotomy into hard and easy problems.

A Dichotomy for First-Order Reducts of Unary Structures

- Mathematics, Computer ScienceLog. Methods Comput. Sci.
- 2018

This work uses a general polynomial-time reduction from such infinite-domain CSPs to finite- domains to obtain new powerful polynometric-time tractability conditions that can be expressed in terms of the topological polymorphism clone of A.