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" of the hierarchy, and a geometric part -- which we call tensorisation -- inspired by multilinear algebra. We show that the hierarchies of minion tests obtained in this way are general enough to capture the (combinatorial) bounded width and also the Sherali-Adams LP, Sum-of-Squares SDP, and affine IP hierarchies. We exploit the geometry of the tensor spaces arising from our construction to prove general properties of such hierarchies. We identify certain classes of minions, which we call linear and conic, whose corresponding hierarchies have particularly fine features. Finally, 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:我们提供了一个统一的框架,用于研究约束性满意度问题及其承诺变体的放松等级结构。 我们的想法是将等级描述分为代数部分,这取决于一个抓住等级“基准”的迷你符和几何部分 -- -- 我们称之为推力 -- -- 受多线性代数的启发。 我们显示,以这种方式获得的迷你试验的等级结构非常笼统,足以捕捉(combinatory)约束的宽度,以及Sherali-Adams LP、Sum- of Squares SDP 和affine IP等级。我们利用建筑中产生的高温空间的几何方法来证明这种等级的一般特性。 我们确定了某些微小的类别,我们称之为线性和共性,其相应的等级特别细。 最后,为了分析Sum-squares SDP等级,我们还将标准SDP放松的软性通过一个新的迷你来描述。