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来研究约束满足问题及其承诺变体的松弛层级结构。该框架的核心思想是将层级结构的描述分为两部分：代数部分依赖于捕获"基础层级"的minion，几何部分——我们称之为张量化——则受到多重线性代数的启发。我们利用构造产生的张量空间的几何性质来证明层级结构的普遍特性。我们识别出特定类别的minion（称为线性和锥性minion），其对应层级结构具有特别优良的特性。我们证实了（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"线性规划+仿射整数规划"组合层级均可被此框架所描述。特别地，为分析平方和半定规划层级，我们还通过新型minion刻画了标准半定规划松弛的可解性。