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作为约束满足问题（CSPs）及其承诺版本之间的归约工具，旨在以类似于代数方法对CSP间构件归约进行分类的方式，对这些归约进行分类。这项研究源于承诺CSP领域中对更具表达力的归约形式的需求。虽然构件归约足以提供（有限域）非承诺CSP中的所有必要困难性，但在承诺CSP中需要采用更广泛的归约类别。我们提出一个通用的归约框架，称为一致性归约，该框架涵盖了最近用于证明承诺CSP的NP困难性的大多数（即使不是全部）归约方法。我们证明了这些归约的一些基本性质，并通过用模板的多态性刻画与弧一致性相关的片段，为理解一致性归约的能力迈出了第一步。除展示困难性外，一致性归约还可通过归约到固定的可解（承诺）CSP（例如求解仿射方程组）来提供可行算法。在此方向上，除其他结果外，我们还用线性规划的一致性归约描述了著名的CSP Sherali-Adams层级体系。