We provide a unified framework to study hierarchies of relaxations for Constraint Satisfaction Problems and their Promise variant. The idea is to split the description of a hierarchy into an algebraic part, depending on a minion capturing the "base level", and a geometric part - which we call tensorisation - inspired by multilinear algebra. We exploit the geometry of the tensor spaces arising from our construction to prove general properties of hierarchies. We identify certain classes of minions, which we call linear and conic, whose corresponding hierarchies have particularly fine features. We establish that the (combinatorial) bounded width, Sherali-Adams LP, affine IP, Sum-of-Squares SDP, and combined "LP + affine IP" hierarchies are all captured by this framework. In particular, in order to analyse the Sum-of-Squares SDP hierarchy, we also characterise the solvability of the standard SDP relaxation through a new minion.

翻译：我们提出了一个统一框架，用于研究约束满足问题及其承诺变体的松弛层次结构。该框架将层次结构的描述拆分为两部分：基于捕获“基级”的极小元的代数部分，以及受多重线性代数启发的几何部分（称为张量化）。我们利用构造中出现的张量空间的几何特性来证明层次结构的一般性质。我们识别出两类极小元（称为线性极小元和圆锥极小元），其对应的层次结构具有特别精细的特征。我们证明了（组合）有界宽度、Sherali-Adams线性规划、仿射整数规划、平方和半定规划以及组合“线性规划+仿射整数规划”层次结构均可纳入此框架。特别地，为了分析平方和半定规划层次结构，我们还通过一种新型极小元刻画了标准半定规划松弛的可解性。