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Singleton algorithms for the Constraint Satisfaction Problem

Dmitriy Zhuk

Abstract

A natural strengthening of an algorithm for the (promise) constraint satisfaction problem is its singleton version: we first fix a constraint to a tuple from the constraint relation, then run the algorithm, and remove the tuple from the constraint if the answer is negative. We characterize the power of the singleton versions of standard universal algorithms for the (promise) CSP over a fixed template in terms of the existence of a minion homomorphism. Using the Hales-Jewett theorem, we show that for finite relational structures this minion condition is equivalent to the existence of polymorphisms with certain symmetries, which we call palette symmetric polymorphisms. By proving the existence of such polymorphisms we establish that the singleton version of the BLP+AIP algorithm solves all (multi-sorted) tractable CSPs over domains of size at most 7. We further show that already for domain size 8 there exists a relational structure arising from the dihedral group $D_4$ that does not admit palette symmetric polymorphisms and cannot be solved by singleton BLP+AIP. By providing concrete CSP templates, we illustrate the limitations of linear programming, the power of the singleton versions, and the elegance of palette symmetric polymorphisms. Surprisingly, we also identify a concrete temporal relational structure whose CSP can be solved by a singleton algorithm after sampling, yet it does not admit palette symmetric polymorphisms. This highlights that the Hales-Jewett argument applies only to finite structures. Nevertheless, we introduce generalized palette polymorphisms for every tractable temporal relational structure and show that they yield a rounding procedure implying that the corresponding singleton algorithm solves its CSP after sampling.

Singleton algorithms for the Constraint Satisfaction Problem

Abstract

A natural strengthening of an algorithm for the (promise) constraint satisfaction problem is its singleton version: we first fix a constraint to a tuple from the constraint relation, then run the algorithm, and remove the tuple from the constraint if the answer is negative. We characterize the power of the singleton versions of standard universal algorithms for the (promise) CSP over a fixed template in terms of the existence of a minion homomorphism. Using the Hales-Jewett theorem, we show that for finite relational structures this minion condition is equivalent to the existence of polymorphisms with certain symmetries, which we call palette symmetric polymorphisms. By proving the existence of such polymorphisms we establish that the singleton version of the BLP+AIP algorithm solves all (multi-sorted) tractable CSPs over domains of size at most 7. We further show that already for domain size 8 there exists a relational structure arising from the dihedral group that does not admit palette symmetric polymorphisms and cannot be solved by singleton BLP+AIP. By providing concrete CSP templates, we illustrate the limitations of linear programming, the power of the singleton versions, and the elegance of palette symmetric polymorphisms. Surprisingly, we also identify a concrete temporal relational structure whose CSP can be solved by a singleton algorithm after sampling, yet it does not admit palette symmetric polymorphisms. This highlights that the Hales-Jewett argument applies only to finite structures. Nevertheless, we introduce generalized palette polymorphisms for every tractable temporal relational structure and show that they yield a rounding procedure implying that the corresponding singleton algorithm solves its CSP after sampling.

Paper Structure

This paper contains 38 sections, 88 theorems, 53 equations, 2 tables, 3 algorithms.

Table of Contents

  1. Introduction
  2. Constraint Satisfaction Problem
  3. Universal algorithms and their singleton versions
  4. Main results
  5. Structure of the paper
  6. Preliminaries
  7. Universal algorithms
  8. Arc-consistency algorithm
  9. Linear relaxations
  10. Combinations of the algorithms
  11. Singleton algorithms and palette functions
  12. Limitations of linear programming
  13. Classic templates
  14. Disconnected templates
  15. Implications of linear equations
  16. ...and 23 more sections

Key Result

Theorem 1.1

Suppose $\mathbb A$ is a finite relational structure. Then $\mathop{\mathrm{CSP}}\nolimits(\mathbb A)$ is solvable in polynomial time if there exists a WNU polymorphism of $\mathbb A$; $\mathop{\mathrm{CSP}}\nolimits(\mathbb A)$ is NP-complete otherwise.

Theorems & Definitions (171)

  • Theorem 1.1: zhuk2020proofZhukFVConjectureBulatovFVConjectureBulatovProofCSP
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 3.1: Theorem 7.4 in PCSPBible
  • Theorem 3.2: Theorem 7.9 in PCSPBible
  • Theorem 3.3: Theorem 7.19 in PCSPBible
  • Theorem 3.4: Theorem 5.1 in BLPplusAIP
  • Remark 3.1
  • Theorem 3.5
  • Theorem 3.6
  • ...and 161 more