# Classifying the Complexity of Constraints Using Finite Algebras

@article{Bulatov2005ClassifyingTC, title={Classifying the Complexity of Constraints Using Finite Algebras}, author={Andrei A. Bulatov and Peter Jeavons and Andrei A. Krokhin}, journal={SIAM J. Comput.}, year={2005}, volume={34}, pages={720-742} }

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. Here we show that any set of relations used to specify the allowed forms of constraints can be associated with a finite universal algebra and we explore how the computational complexity of the corresponding constraint satisfaction problem is connected to the…

## 569 Citations

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- 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.

### An Algebraic Characterisation of Complexity for Valued Constraint

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It is shown that the existence of a non-trivial fractional polymorphism is a necessary condition for the tractability of a valued constraint language with rational or infinite costs over any finite domain (assuming P ≠ NP).

### Constraint Satisfaction Problems over Finite Structures

- Mathematics2021 36th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS)
- 2021

It is shown that every finite equationally nontrivial algebra has this property which gives, as a simple consequence, a complete complexity classification of CSPs over two-element structures, thus extending the classification for two- element relational structures by Schaefer (STOC’78).

### Binarisation via Dualisation for Valued Constraints

- Computer ScienceAAAI
- 2015

A simple proof of the fact that to classify the computational complexity of allvalued constraint languages it suffices to classify only binary valued constraint languages.

### An Algorithm for Constraint Satisfaction Problem

- Computer Science, Mathematics2017 IEEE 47th International Symposium on Multiple-Valued Logic (ISMVL)
- 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.

### Proof Complexity Meets Algebra

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It is shown that, for the most studied propositional, algebraic, and semialgebraic proof systems, the classical constructions of pp-interpretability, homomorphic equivalence, and addition of constants to a core preserve the proof complexity of the CSP.

### 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.

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