# Universal Algebraic Methods for Constraint Satisfaction Problems

@article{Bergman2022UniversalAM, title={Universal Algebraic Methods for Constraint Satisfaction Problems}, author={Clifford Bergman and William DeMeo}, journal={ArXiv}, year={2022}, volume={abs/1611.02867} }

After substantial progress over the last 15 years, the "algebraic
CSP-dichotomy conjecture" reduces to the following: every local constraint
satisfaction problem (CSP) associated with a finite idempotent algebra is
tractable if and only if the algebra has a Taylor term operation. Despite the
tremendous achievements in this area (including recently announce proofs of the
general conjecture), there remain examples of small algebras with just a single
binary operation whose CSP resists direct…

## One Citation

### Algebraic Approach to Constraint Satisfaction Problems

- Mathematics
- 2016

• Suppose B is a proper subalgebra of A, and C is a proper subalgebra of B. If C is absorbing in A, then C is absorbing in B. • Every finite abelian algebra is absorption-free. • If B is absorbing in…

## References

SHOWING 1-10 OF 33 REFERENCES

### Universal Algebraic Methods for Constraint Satisfaction Problems Part I. Preliminaries

- Mathematics, Computer Science

This paper presents some new methods for approaching the class of finite algebras known as " commutative idempotent binars " (CIBs) and demonstrates the utility of these methods by using them to prove that every CIB of cardinality at most 4 yields a tractable CSP.

### 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.

### 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 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.

### Absorbing Subalgebras, Cyclic Terms, and the Constraint Satisfaction Problem

- MathematicsLog. Methods Comput. Sci.
- 2012

Two new characterizations of finitely generated Taylor varieties are provided using absorbing subalgebras and cyclic terms, allowing to reprove the conjecture of Bang-Jensen and Hell and the characterization of locally finite Taylor varieties using weak near- unanimity terms in an elementary and self-contained way.

### Varieties with few subalgebras of powers

- Mathematics
- 2009

The Constraint Satisfaction Problem Dichotomy Conjecture of Feder and Vardi (1999) has in the last 10 years been profitably reformulated as a conjecture about the set SP fin (A) of subalgebras of…

### 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 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.

### Absorption in Universal Algebra and CSP

- Computer ScienceThe Constraint Satisfaction Problem
- 2017

The concept of absorption is introduced, its use in a number of basic proofs is illustrated and an overview of the most important results obtained by using it is provided.

### 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.