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Local consistency as a reduction between constraint satisfaction problems

Victor Dalmau, Jakub Opršal

TL;DR

This paper develops a general framework of consistency reductions for (promise) CSPs, unifying gadget reductions within Datalog-based interpretations. It proves that Datalog reductions and $k$-consistency reductions have the same power and links them to hierarchies such as bounded width and Sherali–Adams, while providing an arc-consistency characterisation via a minion transformation on polymorphisms. It extends to hierarchies over Abelian groups and affine systems of equations, and discusses conjectures that all tractable finite-template CSPs reduce to affine equations over $\oldsymbol{Z}$ under Datalog reductions. These results offer a cohesive algebraic/combinatorial framework for tractability and hardness in promise CSPs and point toward a unifying theory for reductions in this broader landscape.

Abstract

We study the use of local consistency methods as reductions between constraint satisfaction problems (CSPs), and promise version thereof, with the aim to classify these reductions in a similar way as the algebraic approach classifies gadget reductions between CSPs. This research is motivated by the requirement of more expressive reductions in the scope of promise CSPs. While gadget reductions are enough to provide all necessary hardness in the scope of (finite domain) non-promise CSP, in promise CSPs a wider class of reductions needs to be used. We provide a general framework of reductions, which we call consistency reductions, that covers most (if not all) reductions recently used for proving NP-hardness of promise CSPs. We prove some basic properties of these reductions, and provide the first steps towards understanding the power of consistency reductions by characterizing a fragment associated to arc-consistency in terms of polymorphisms of the template. In addition to showing hardness, consistency reductions can also be used to provide feasible algorithms by reducing to a fixed tractable (promise) CSP, for example, to solving systems of affine equations. In this direction, among other results, we describe the well-known Sherali-Adams hierarchy for CSP in terms of a consistency reduction to linear programming.

Local consistency as a reduction between constraint satisfaction problems

TL;DR

This paper develops a general framework of consistency reductions for (promise) CSPs, unifying gadget reductions within Datalog-based interpretations. It proves that Datalog reductions and -consistency reductions have the same power and links them to hierarchies such as bounded width and Sherali–Adams, while providing an arc-consistency characterisation via a minion transformation on polymorphisms. It extends to hierarchies over Abelian groups and affine systems of equations, and discusses conjectures that all tractable finite-template CSPs reduce to affine equations over under Datalog reductions. These results offer a cohesive algebraic/combinatorial framework for tractability and hardness in promise CSPs and point toward a unifying theory for reductions in this broader landscape.

Abstract

We study the use of local consistency methods as reductions between constraint satisfaction problems (CSPs), and promise version thereof, with the aim to classify these reductions in a similar way as the algebraic approach classifies gadget reductions between CSPs. This research is motivated by the requirement of more expressive reductions in the scope of promise CSPs. While gadget reductions are enough to provide all necessary hardness in the scope of (finite domain) non-promise CSP, in promise CSPs a wider class of reductions needs to be used. We provide a general framework of reductions, which we call consistency reductions, that covers most (if not all) reductions recently used for proving NP-hardness of promise CSPs. We prove some basic properties of these reductions, and provide the first steps towards understanding the power of consistency reductions by characterizing a fragment associated to arc-consistency in terms of polymorphisms of the template. In addition to showing hardness, consistency reductions can also be used to provide feasible algorithms by reducing to a fixed tractable (promise) CSP, for example, to solving systems of affine equations. In this direction, among other results, we describe the well-known Sherali-Adams hierarchy for CSP in terms of a consistency reduction to linear programming.
Paper Structure (34 sections, 41 theorems, 87 equations, 1 figure)

This paper contains 34 sections, 41 theorems, 87 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Constraint satisfaction problems
  4. Reductions
  5. Gadget reductions
  6. Local consistency reductions
  7. Logical reductions
  8. Combinatorial reductions
  9. Hierarchies and CSP algorithms
  10. Bounded width
  11. Sherali--Adams
  12. Hierarchies of groups
  13. Characterisation of the arc-consistency reduction
  14. Conclusion
  15. Preliminaries to the appendices
  16. ...and 19 more sections

Key Result

Theorem 1

Every gadget reduction is expressible as a Datalog reduction up to homomorphic equivalence.

Figures (1)

  • Figure 1: Boolean CSPs ordered by three classes of reductions with examples of problems belonging to some classes.

Theorems & Definitions (100)

  • Theorem 1: Theorem \ref{['thm:gadget-is-ddatalog']} informally
  • Theorem 2: Theorem \ref{['thm:canonical-width']} informally
  • Conjecture 1: Conjecture \ref{['the-conjecture']}
  • Theorem 3: Theorem \ref{['thm:unary']} informally
  • definition 1
  • definition 2
  • definition 3
  • theorem 1: Barto, Bulín, Krokhin, and Opršal BBKO21
  • definition 4
  • lemma 1
  • ...and 90 more