The complexity of the promise constraint satisfaction problem $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$ is largely unknown, even for symmetric $\mathbf{A}$ and $\mathbf{B}$, except for the case when $\mathbf{A}$ and $\mathbf{B}$ are Boolean. First, we establish a dichotomy for $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$ where $\mathbf{A}, \mathbf{B}$ are symmetric, $\mathbf{B}$ is functional (i.e. any $r-1$ elements of an $r$-ary tuple uniquely determines the last one), and $(\mathbf{A},\mathbf{B})$ satisfies technical conditions we introduce called dependency and additivity. This result implies a dichotomy for $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$ with $\mathbf{A},\mathbf{B}$ symmetric and $\mathbf{B}$ functional if (i) $\mathbf{A}$ is Boolean, or (ii) $\mathbf{A}$ is a hypergraph of a small uniformity, or (iii) $\mathbf{A}$ has a relation $R^{\mathbf{A}}$ of arity at least 3 such that the hypergraph diameter of $(A, R^{\mathbf{A}})$ is at most 1. Second, we show that for $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$, where $\mathbf{A}$ and $\mathbf{B}$ contain a single relation, $\mathbf{A}$ satisfies a technical condition called balancedness, and $\mathbf{B}$ is arbitrary, the combined basic linear programming relaxation (BLP) and the affine integer programming relaxation (AIP) is no more powerful than the (in general strictly weaker) AIP relaxation. Balanced $\mathbf{A}$ include symmetric $\mathbf{A}$ or, more generally, $\mathbf{A}$ preserved by a transitive permutation group.

翻译：承诺约束满足问题 $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$ 的复杂度在很大程度上仍是未知的，即便在 $\mathbf{A}$ 和 $\mathbf{B}$ 均为对称结构的情况下，除二者为布尔结构的情形外仍如此。首先，我们为 $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$ 建立了一个二分法定理，其中 $\mathbf{A},\mathbf{B}$ 均为对称结构，$\mathbf{B}$ 是函数型的（即 $r$ 元元组中任意 $r-1$ 个元素可唯一确定最后一个元素），且 $(\mathbf{A},\mathbf{B})$ 满足我们引入的称为依赖性和可加性的技术条件。该结果蕴含了在以下三种情形下 $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$ 的二分法：（i）$\mathbf{A}$ 为布尔结构；（ii）$\mathbf{A}$ 是一致性较小的超图；（iii）$\mathbf{A}$ 包含一个元数至少为3的关系 $R^{\mathbf{A}}$，使得 $(A, R^{\mathbf{A}})$ 的超图直径不超过1。其次，我们证明对于 $\operatorname{PCSP}(\mathbf{A},\mathbf{B})$，其中 $\mathbf{A}$ 和 $\mathbf{B}$ 仅包含单一关系，$\mathbf{A}$ 满足称为平衡性的技术条件，且 $\mathbf{B}$ 为任意结构，此时联合基本线性规划松弛（BLP）与仿射整数规划松弛（AIP）的威力并不超过（通常严格更弱的）AIP松弛。平衡的 $\mathbf{A}$ 包括对称结构 $\mathbf{A}$，或更一般地，那些被可迁置换群所保持的 $\mathbf{A}$。