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Algebraic and algorithmic synergies between promise and infinite-domain CSPs

Antoine Mottet

TL;DR

This work builds a unifying bridge between infinite-domain CSPs and promise CSPs by leveraging algebraic and categorical tools. It shows that CSPs in the Bodirsky-Pinsker class are polynomial-time equivalent to half-infinite PCSPs with an infinite left template and a finite right template, and develops uniform polynomial-time algorithms for temporal CSPs that certify unsatisfiability when appropriate. It introduces and analyzes the $\Gamma_m$ construction, adjunctions, and minion homomorphisms to transfer polymorphism identities and tractability criteria across CSP/PCSP templates, deriving both hardness criteria (Olšák, Siggers, WNU modulo automorphisms) and tractability results. Importantly, the paper constructs finite PCSP templates that have width $4$ and possess omega-categorical tractable sandwiches but are not finitely tractable and not solvable by BLP+AIP, revealing a nuanced interplay between finite and infinite CSPs and clarifying the landscape of tractability for PCSPs.

Abstract

We establish a framework that allows us to transfer results between some constraint satisfaction problems with infinite templates and promise constraint satisfaction problems. On the one hand, we obtain new algebraic results for infinite-domain CSPs giving new criteria for NP-hardness. On the other hand, we show the existence of promise CSPs with finite templates that reduce naturally to tractable infinite-domain CSPs within the scope of the Bodirsky-Pinsker conjecture, but that are not finitely tractable, thereby showing a non-trivial connection between those two fields of research. In an important part of our proof, we also obtain uniform polynomial-time algorithms solving temporal constraint satisfaction problems.

Algebraic and algorithmic synergies between promise and infinite-domain CSPs

TL;DR

This work builds a unifying bridge between infinite-domain CSPs and promise CSPs by leveraging algebraic and categorical tools. It shows that CSPs in the Bodirsky-Pinsker class are polynomial-time equivalent to half-infinite PCSPs with an infinite left template and a finite right template, and develops uniform polynomial-time algorithms for temporal CSPs that certify unsatisfiability when appropriate. It introduces and analyzes the construction, adjunctions, and minion homomorphisms to transfer polymorphism identities and tractability criteria across CSP/PCSP templates, deriving both hardness criteria (Olšák, Siggers, WNU modulo automorphisms) and tractability results. Importantly, the paper constructs finite PCSP templates that have width and possess omega-categorical tractable sandwiches but are not finitely tractable and not solvable by BLP+AIP, revealing a nuanced interplay between finite and infinite CSPs and clarifying the landscape of tractability for PCSPs.

Abstract

We establish a framework that allows us to transfer results between some constraint satisfaction problems with infinite templates and promise constraint satisfaction problems. On the one hand, we obtain new algebraic results for infinite-domain CSPs giving new criteria for NP-hardness. On the other hand, we show the existence of promise CSPs with finite templates that reduce naturally to tractable infinite-domain CSPs within the scope of the Bodirsky-Pinsker conjecture, but that are not finitely tractable, thereby showing a non-trivial connection between those two fields of research. In an important part of our proof, we also obtain uniform polynomial-time algorithms solving temporal constraint satisfaction problems.
Paper Structure (27 sections, 54 theorems, 23 equations, 1 figure)

This paper contains 27 sections, 54 theorems, 23 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Tractability by infinite CSPs
  3. Uniform Algorithms for Temporal CSPs
  4. Identities
  5. Definitions
  6. Countably categorical structures
  7. Polymorphisms
  8. Algorithmic notions
  9. Local consistency
  10. Relaxations
  11. Uniform Algorithms for Temporal Constraints
  12. Templates with $pp$
  13. Templates with $ll$
  14. Conclusion
  15. PCSP templates from CSP templates
  16. ...and 12 more sections

Key Result

theorem 1

Every problem $\mathop{\mathrm{CSP}}\nolimits(\mathbb{C})$ in the scope of the Bodirsky-Pinsker conjecture is polynomial-time equivalent to a problem of the form $\mathop{\mathrm{PCSP}}\nolimits(\mathbb{A},\mathbb{B})$, where $\mathbb{A}$ is in the scope of the Bodirsky-Pinsker conjecture, and $\mat

Figures (1)

  • Figure 1: Diagram representing the various arrows in \ref{['directedsystem', 'proj-limit']}. Solid arrows are local minion homomorphisms, and the dotted arrow is a $(*,\overline{\mathscr G})$-homomorphism.

Theorems & Definitions (100)

  • theorem 1
  • theorem 2
  • theorem 3
  • theorem 4
  • theorem 5
  • lemma 5
  • definition 6: Similarly as TemporalDescriptive
  • lemma 6
  • lemma 7
  • proof
  • ...and 90 more