# Constraint Satisfaction Problems over Finite Structures

@article{Barto2021ConstraintSP, title={Constraint Satisfaction Problems over Finite Structures}, author={Libor Barto and William DeMeo and Antoine Mottet}, journal={2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)}, year={2021}, pages={1-13} }

We initiate a systematic study of the computational complexity of the Constraint Satisfaction Problem (CSP) over finite structures that may contain both relations and operations. We show the close connection between this problem and a natural algebraic question: which finite algebras admit only polynomially many homomorphisms into them?We give some sufficient and some necessary conditions for a finite algebra to have this property. In particular, we show that every finite equationally…

## References

SHOWING 1-10 OF 40 REFERENCES

Classifying the Complexity of Constraints Using Finite Algebras

- Mathematics, Computer ScienceSIAM 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.

Constraint Satisfaction with Countable Homogeneous Templates

- Computer Science, MathematicsJ. Log. Comput.
- 2006

It is proved that the primitive positive definable relations over an ω-categorical structure Γ are precisely the relations that are preserved by the polymorphisms of Γ.

The wonderland of reflections

- Mathematics, Computer ScienceArXiv
- 2015

A new elegant dichotomy conjecture for the CSPs of reducts of finitely bounded homogeneous structures is formulated and a close connection between h1 clone homomorphisms and the notion of compatibility with projections used in the study of the lattice of interpretability types of varieties is revealed.

A Proof of CSP Dichotomy Conjecture

- Computer Science, Mathematics2017 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.

On the Algebraic Structure of Combinatorial Problems

- Computer Science, MathematicsTheor. Comput. Sci.
- 1998

A general algebraic formulation for a wide range of combinatorial problems including Satisfiability, Graph Colorability and Graph Isomorphism is described, and it is demonstrated that the complexity of solving this decision problem is determined in many cases by simple algebraic properties of the relational structures involved.

Dichotomies for classes of homomorphism problems involving unary functions

- Computer Science, MathematicsTheor. Comput. Sci.
- 2004

A dichotomy result is established for the class of non-uniform constraint satisfaction problems where the template is a λ2-structure with the property that the two unary functions involved are the reverse of one another, in that every such problem is either solvable in polynomial-time or NP-complete.

Algebraic approach to promise constraint satisfaction

- Computer Science, MathematicsSTOC
- 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 universal-algebraic proof of the complexity dichotomy for Monotone Monadic SNP

- Computer Science, MathematicsLICS
- 2018

This work presents a new proof of the reduction to finite-domain CSPs that does not rely on the results of Kun, and uses the universal-algebraic approach to study the computational complexity of MMSNP.

The Computational Structure of Monotone Monadic SNP and Constraint Satisfaction: A Study through Datalog and Group Theory

- Mathematics, Computer ScienceSIAM J. Comput.
- 1998

This paper isolates a class (of problems specified by) "monotone monadic SNP without inequality" which may exhibit a dichotomy, and explains the placing of all these restrictions by showing, essentially using Ladner's theorem, that classes obtained by using only two of the above three restrictions do not show this dichotomy.

The complexity of the counting constraint satisfaction problem

- Computer Science, MathematicsJACM
- 2013

This article characterize relational structures H for which (#CSP(H) can be solved in polynomial time and prove that for all other structures the problem is #P-complete.