一句话：证明布尔对称模板对函数式目标 PCSP 的二分性，并比较 BLP+AIP 与 AIP 的强度。
本文研究 promise 约束满足问题 PCSP(A,B)。第一项结果：若 A 是布尔且对称，B 是函数式（任意 domain；任意关系中除一个元素外，其余元素决定最后一个元素），则 PCSP(A,B) 具有二分性。该结果涵盖 B 函数式时的 PCSP(q-in-r,B)，并推进了 PCSP(1-in-3,B) 的分类（Barto 等此前仅处理三元素 domain 的 B）。第二项结果：若 A 只含一个对称关系，而 B 任意（不必函数式），则 Brakensiek 等提出的 BLP 与 AIP 联合松弛不比 Brakensiek 和 Guruswami 的 AIP 松弛更强，后者一般严格更弱。
As our first result, we establish a dichotomy for promise constraint satisfaction problems of the form PCSP(A, B), where A is Boolean and symmetric and B is functional (on a domain of any size); i.e, all but one element of any tuple in a relation in B determine the last element. This includes PCSPs of the form PCSP(q-in-r, B), where B is functional, thus making progress towards a classification of PCSP(1-in-3, B), which were studied by Barto, Battistelli, and Berg [STACS’21] for B on three-element domains.As our second result, we show that for PCSP(A, B), where A contains a single symmetric relation and B is arbitrary (and thus not necessarily functional), the combined basic linear programming relaxation (BLP) and the affine integer programming relaxation (AIP) of Brakensiek et al. [SICOMP’20] is no more powerful than the (in general strictly weaker) AIP relaxation of Brakensiek and Guruswami [SICOMP’21].