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A simplified proof of the CSP Dichotomy Conjecture and XY-symmetric operations

Dmitriy Zhuk

TL;DR

A new theory of strong subalgebras and linear congruences that are defined globally is developed and it is proved that composing a weak near-unanimity operation of an odd arity $n$ can derive an symmetric operation that is symmetric on all two-element sets.

Abstract

We develop a new theory of strong subalgebras and linear congruences that are defined globally. Using this theory we provide a new proof of the correctness of Zhuk's algorithm for all tractable CSPs on a finite domain, and therefore a new simplified proof of the CSP Dichotomy Conjecture. Additionally, using the new theory we prove that composing a weak near-unanimity operation of an odd arity $n$ we can derive an $n$-ary operation that is symmetric on all two-element sets. Thus, CSP over a constraint language $Γ$ on a finite domain is tractable if and only if there exist infinitely many polymorphisms of $Γ$ that are symmetric on all two-element sets.

A simplified proof of the CSP Dichotomy Conjecture and XY-symmetric operations

TL;DR

A new theory of strong subalgebras and linear congruences that are defined globally is developed and it is proved that composing a weak near-unanimity operation of an odd arity can derive an symmetric operation that is symmetric on all two-element sets.

Abstract

We develop a new theory of strong subalgebras and linear congruences that are defined globally. Using this theory we provide a new proof of the correctness of Zhuk's algorithm for all tractable CSPs on a finite domain, and therefore a new simplified proof of the CSP Dichotomy Conjecture. Additionally, using the new theory we prove that composing a weak near-unanimity operation of an odd arity we can derive an -ary operation that is symmetric on all two-element sets. Thus, CSP over a constraint language on a finite domain is tractable if and only if there exist infinitely many polymorphisms of that are symmetric on all two-element sets.
Paper Structure (27 sections, 93 theorems, 57 equations, 2 algorithms)

This paper contains 27 sections, 93 theorems, 57 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. A simplified proof of the CSP Dichotomy Conjecture
  3. Existence of XY-symmetric operations
  4. History and acknowledgements
  5. Structure of the paper
  6. Strong/Linear subuniverses
  7. Auxiliary definitions
  8. Definition of strong subuniverses
  9. Properties of strong subuniverses
  10. Proof of the CSP Dichotomy Conjecture
  11. Additional definitions
  12. Auxiliary statements
  13. Main Statements
  14. Statements sufficient to prove that Zhuk's algorithm works
  15. XY-symmetric operations
  16. ...and 12 more sections

Key Result

Theorem 1

Suppose $\Gamma$ is a finite set of relations on a finite set $A$. Then $\mathop{\mathrm{CSP}}\nolimits(\Gamma)$ can be solved in polynomial time if there exists a WNU preserving $\Gamma$; $\mathop{\mathrm{CSP}}\nolimits(\Gamma)$ is NP-complete otherwise.

Theorems & Definitions (169)

  • Theorem 1: zhuk2020proofZhukFVConjectureBulatovFVConjectureBulatovProofCSP
  • Theorem 2
  • Corollary 3
  • Lemma 4
  • proof
  • Lemma 5
  • Lemma 6: zhuk2021strong, Corollary 6.11.1
  • Lemma 7
  • Lemma 8
  • Lemma 9
  • ...and 159 more