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.

翻译：我们研究局部一致性方法在约束满足问题（CSPs）及其承诺版本中作为约简手段的应用，旨在以类似于代数方法分类CSPs间小工具约简的方式对这些约简进行分类。这一研究源于承诺CSPs领域对更具表达能力的约简方法的需求。虽然小工具约简足以在（有限域）非承诺CSP中提供所有必要的困难性，但在承诺CSP中需要使用更广泛的约简类别。我们提出了一个称为一致性约简的通用约简框架，该框架涵盖了最近用于证明承诺CSPs NP困难性的大部分（若非全部）约简方法。我们证明了这些约简的基本性质，并通过基于模板的多态性刻画与弧一致性相关的片段，迈出了理解一致性约简能力的第一步。除展示困难性外，一致性约简还可通过将问题约简为固定的易解（承诺）CSP（例如求解仿射方程组）来提供可行算法。在此方向上，除其他成果外，我们以线性规划的一致性约简形式描述了著名的Sherali-Adams层次结构。