# A Proof of the CSP Dichotomy Conjecture

## TL;DR

This paper proves the CSP Dichotomy Conjecture by showing that constraint languages with a weak near unanimity polymorphism are solvable in polynomial time, establishing a complete classification of CSP complexity.

## Contribution

It provides the first polynomial-time algorithm for CSPs with weak near unanimity polymorphisms, confirming the conjecture's remaining part.

## Key findings

- Polynomial-time algorithm for CSPs with weak near unanimity polymorphisms
- Complete classification of CSP complexity based on polymorphisms
- Resolution of the long-standing CSP Dichotomy Conjecture

## Abstract

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 conjectured that if a constraint language has a weak near unanimity polymorphism then the corresponding constraint satisfaction problem is tractable, otherwise it is NP-complete. In the paper we present 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.