一句话：提出基于平滑分层 PCP 定理的单射硬度条件，据此证明一类 PCSP 模板 NP 难，并建立布尔线性阈值模板的二分定理。
本文研究 Promise 约束满足问题（PCSP）的硬度判定。现有最先进的硬度条件（ACMTCT'21）并不完备：作者构造了一个 PCSP 模板，它已知是 NP 难的，却不满足该条件。为此，他们基于分层 PCP 定理的平滑版本提出新的“单射”硬度条件，并用它证明该模板确实 NP 难。文章第二部分进一步对由线性阈值函数作为多态运算定义的布尔 PCSP 模板进行完整分类，给出二分定理；该分类的推理依赖于这一新的单射条件。这些结果扩展了 PCSP 硬度判定的工具。
We present a template for the Promise Constraint Satisfaction Problem (PCSP) which is NP-hard but does not satisfy the current state-of-the-art hardness condition [ACMTCT'21]. We introduce a new "injective" condition based on the smooth version of the layered PCP Theorem and use this new condition to confirm that the problem is indeed NP-hard. In the second part of the article, we establish a dichotomy for Boolean PCSPs defined by templates with polymorphisms in the set of linear threshold functions. The reasoning relies on the new injective condition.