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Strict width for Constraint Satisfaction Problems over homogeneous strucures of finite duality

Tomáš Nagy, Michael Pinsker

TL;DR

The paper analyzes when the local-consistency principle implies global solvability for infinite-domain CSP templates within the Bodirsky-Pinsker framework. It develops an implicational framework for FO expansions of $k$-neoliberal, finitely bounded homogeneous ground structures with finite duality and connects bounded strict width to implicational-simplicity, yielding an explicit relational width bound $ (k,\text{max}(k+1,b_{oldsymbol{B}})) $. The approach uses injective reductions, implication graphs, and the algebraic properties of polymorphisms to extend prior binary-restricted results to Hypergraph-SAT and broader template families. These results advance the program of classifying infinite-domain CSPs by structural identities and provide concrete, computable local-consistency thresholds for broad classes of problems with potential algorithmic impact.

Abstract

We investigate the `local consistency implies global consistency' principle of strict width among structures within the scope of the Bodirsky-Pinsker dichotomy conjecture for infinite-domain Constraint Satisfaction Problems (CSPs). Our main result implies that for certain CSP templates within the scope of that conjecture, having bounded strict width has a concrete consequence on the expressive power of the template called implicational simplicity. This in turn yields an explicit bound on the relational width of the CSP, i.e., the amount of local consistency needed to ensure the satisfiability of any instance. Our result applies to first-order expansions of any homogeneous $k$-uniform hypergraph, but more generally to any CSP template under the assumption of finite duality and general abstract conditions mainly on its automorphism group. In particular, it overcomes the restriction to binary signatures in the pioneering work of Wrona.

Strict width for Constraint Satisfaction Problems over homogeneous strucures of finite duality

TL;DR

The paper analyzes when the local-consistency principle implies global solvability for infinite-domain CSP templates within the Bodirsky-Pinsker framework. It develops an implicational framework for FO expansions of -neoliberal, finitely bounded homogeneous ground structures with finite duality and connects bounded strict width to implicational-simplicity, yielding an explicit relational width bound . The approach uses injective reductions, implication graphs, and the algebraic properties of polymorphisms to extend prior binary-restricted results to Hypergraph-SAT and broader template families. These results advance the program of classifying infinite-domain CSPs by structural identities and provide concrete, computable local-consistency thresholds for broad classes of problems with potential algorithmic impact.

Abstract

We investigate the `local consistency implies global consistency' principle of strict width among structures within the scope of the Bodirsky-Pinsker dichotomy conjecture for infinite-domain Constraint Satisfaction Problems (CSPs). Our main result implies that for certain CSP templates within the scope of that conjecture, having bounded strict width has a concrete consequence on the expressive power of the template called implicational simplicity. This in turn yields an explicit bound on the relational width of the CSP, i.e., the amount of local consistency needed to ensure the satisfiability of any instance. Our result applies to first-order expansions of any homogeneous -uniform hypergraph, but more generally to any CSP template under the assumption of finite duality and general abstract conditions mainly on its automorphism group. In particular, it overcomes the restriction to binary signatures in the pioneering work of Wrona.
Paper Structure (16 sections, 17 theorems, 11 equations)

This paper contains 16 sections, 17 theorems, 11 equations.

Table of Contents

  1. Introduction
  2. CSPs, polymorphisms, and strict width
  3. Strict width within the Bodirsky-Pinsker conjecture
  4. Related work
  5. Preliminaries
  6. Relational structures and permutation groups
  7. Constraint satisfaction problems and bounded width
  8. Polymorphisms
  9. Proof of the main result
  10. Implicationally simple structures
  11. Binary injections and bounded width
  12. Some implications with no bounded strict width
  13. Critical relations
  14. Composition of implications
  15. Digraphs of implications
  16. ...and 1 more sections

Key Result

Theorem 1.1

Let $k\geq 3$, 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 $\mathbb{A}$ has bou

Theorems & Definitions (47)

  • Theorem 1.1
  • Corollary 1.1
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Theorem 2.7: BodirskyDalmau
  • Definition 3.1
  • ...and 37 more