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New Sufficient Algebraic Conditions for Local Consistency over Homogeneous Structures of Finite Duality

Tomáš Nagy, Michael Pinsker, Michał Wrona

TL;DR

The paper addresses the question of when height-1 Maltsev-type identities imply tractability for infinite-domain CSPs within the Bodirsky-Pinsker framework. It proves that for templates that are first-order expansions of $k$-neoliberal, finitely bounded homogeneous ground structures and are invariant under a chain of quasi Jónsson operations, bounded-width solvability follows, specifically relational width $(k, ext{max}(k+1,b_{ ext B}))$ and polynomial-time CSPs. This is achieved by developing a robust theory of implicational structures, including the notions of implicationally simple vs. hard, implication graphs, and critical relations, and by showing that in this setting one can derive the desired width bounds from height-1 identities. The results extend finite-domain algebraic tractability to a broad infinite-domain class, providing tractability results for Graph-SAT, Hypergraph-SAT, MMSNP/GMSNP templates, and other templates with finite duality, and offering a major step toward a general infinite-domain dichotomy via algebraic conditions on polymorphisms. The work thus contributes a concrete, purely algebraic criterion for tractability in infinite-domain CSPs and strengthens the connection between local-consistency methods and global solvability in a broad template class.

Abstract

The path to the solution of Feder-Vardi dichotomy conjecture by Bulatov and Zhuk led through showing that more and more general algebraic conditions imply polynomial-time algorithms for the finite-domain Constraint Satisfaction Problems (CSPs) whose templates satisfy them. These investigations resulted in the discovery of the appropriate height 1 Maltsev conditions characterizing bounded strict width, bounded width, the applicability of the few-subpowers algorithm, and many others. For problems in the range of the similar Bodirsky-Pinsker conjecture on infinite-domain CSPs, one can only find such a characterization for the notion of bounded strict width, with a proof essentially the same as in the finite case. In this paper, we provide the first non-trivial results showing that certain height 1 Maltsev conditions imply bounded width, and in consequence tractability, for a natural subclass of templates within the Bodirsky-Pinsker conjecture which includes many templates in the literature as well as templates for which no complexity classification is known.

New Sufficient Algebraic Conditions for Local Consistency over Homogeneous Structures of Finite Duality

TL;DR

The paper addresses the question of when height-1 Maltsev-type identities imply tractability for infinite-domain CSPs within the Bodirsky-Pinsker framework. It proves that for templates that are first-order expansions of -neoliberal, finitely bounded homogeneous ground structures and are invariant under a chain of quasi Jónsson operations, bounded-width solvability follows, specifically relational width and polynomial-time CSPs. This is achieved by developing a robust theory of implicational structures, including the notions of implicationally simple vs. hard, implication graphs, and critical relations, and by showing that in this setting one can derive the desired width bounds from height-1 identities. The results extend finite-domain algebraic tractability to a broad infinite-domain class, providing tractability results for Graph-SAT, Hypergraph-SAT, MMSNP/GMSNP templates, and other templates with finite duality, and offering a major step toward a general infinite-domain dichotomy via algebraic conditions on polymorphisms. The work thus contributes a concrete, purely algebraic criterion for tractability in infinite-domain CSPs and strengthens the connection between local-consistency methods and global solvability in a broad template class.

Abstract

The path to the solution of Feder-Vardi dichotomy conjecture by Bulatov and Zhuk led through showing that more and more general algebraic conditions imply polynomial-time algorithms for the finite-domain Constraint Satisfaction Problems (CSPs) whose templates satisfy them. These investigations resulted in the discovery of the appropriate height 1 Maltsev conditions characterizing bounded strict width, bounded width, the applicability of the few-subpowers algorithm, and many others. For problems in the range of the similar Bodirsky-Pinsker conjecture on infinite-domain CSPs, one can only find such a characterization for the notion of bounded strict width, with a proof essentially the same as in the finite case. In this paper, we provide the first non-trivial results showing that certain height 1 Maltsev conditions imply bounded width, and in consequence tractability, for a natural subclass of templates within the Bodirsky-Pinsker conjecture which includes many templates in the literature as well as templates for which no complexity classification is known.

Paper Structure

This paper contains 29 sections, 24 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. The main result
  3. Related work
  4. Preliminaries
  5. Relational structures and permutation groups
  6. Constraint satisfaction problems and bounded width
  7. Polymorphisms
  8. Proof of the main result
  9. Implicationally simple structures
  10. Binary injections and quasi Jónsson operations
  11. Some implications which are not preserved by quasi Jónsson operations
  12. Composition of implications
  13. Digraphs of implications
  14. Critical relations
  15. Proof of Theorem \ref{['thm:implsimple']}
  16. ...and 14 more sections

Key Result

Theorem 9

Let $k\geq 2$, let $\mathbb B$ be $k$-neoliberal, and suppose that $\mathbb B$ has finite duality. Suppose that $\mathbb{A}$ is a CSP template which is an expansion of $\mathbb B$ by first-order definable relations (i.e., Boolean combinations of the relations of $\mathbb B$). If $\mathbb{A}$ is inva

Theorems & Definitions (64)

  • Conjecture 1
  • Conjecture 2
  • Definition 3
  • Definition 4
  • Definition 5
  • Example 6
  • Example 7
  • Example 8
  • Theorem 9
  • Corollary 9
  • ...and 54 more