We develop a unified framework to characterize the power of higher-level algorithms for the constraint satisfaction problem (CSP), such as $k$-consistency, the Sherali-Adams LP hierarchy, and the affine IP hierarchy. As a result, solvability of a fixed-template CSP or, more generally, a Promise CSP by a given level is shown to depend only on the polymorphism minion of the template. Similarly, we obtain a minion-theoretic description of $k$-consistency reductions between Promise CSPs. We introduce a new hierarchy of SDP-like vector relaxations with vectors over $\mathbb Z_{p}$ in which orthogonality is imposed on $k$-tuples of vectors. Surprisingly, this relaxation turns out to be equivalent to the $k$-th level of the AIP-$\mathbb{Z}_p$ relaxation. We show that it solves the CSP of the dihedral group $\mathbf{D}_4$, the smallest CSP that fools the singleton BLP+AIP algorithm. Using this vector representation, we further show that the $p$-th level of the $\mathbb{Z}_p$ relaxation solves linear equations modulo $p^2$.

翻译：我们发展了一个统一框架，用于刻画约束满足问题(CSP)中高层算法的能力，例如$k$-一致性、Sherali-Adams LP层次以及仿射IP层次。结果表明，固定模板CSP或更一般地，给定层次的可满足承诺CSP的可解性仅依赖于模板的多态性小集(minion)。类似地，我们获得了承诺CSP之间$k$-一致性约简的minion理论描述。我们引入了一个新的SDP型向量松弛层级，其中向量取自$\mathbb Z_{p}$，并对$k$元向量组施加正交性条件。令人惊讶的是，该松弛被证明等价于AIP-$\mathbb{Z}_p$松弛的第$k$层。我们证明它能解二面体群$\mathbf{D}_4$的CSP——这是欺骗单例BLP+AIP算法的最小CSP。利用这一向量表示，我们进一步证明了$\mathbb{Z}_p$松弛的第$p$层能解模$p^2$线性方程组。