In this paper, we continue the study of robust satisfiability of promise CSPs (PCSPs), initiated in (Brakensiek, Guruswami, Sandeep, STOC 2023 / Discrete Analysis 2025), and obtain the following results: For the PCSP 1-in-3-SAT vs NAE-SAT with negations, we prove that it is hard, under the Unique Games conjecture (UGC), to satisfy $1-Ω(1/\log (1/ε))$ constraints in a $(1-ε)$-satisfiable instance. This shows that the exponential loss incurred by the BGS algorithm for the case of Alternating-Threshold polymorphisms is necessary, in contrast to the polynomial loss achievable for Majority polymorphisms. For any Boolean PCSP that admits Majority polymorphisms, we give an algorithm satisfying $1-O(\sqrtε)$ fraction of the weaker constraints when promised the existence of an assignment satisfying $1-ε$ fraction of the stronger constraints. This significantly generalizes the Charikar--Makarychev--Makarychev algorithm for 2-SAT, and matches the optimal trade-off possible under the UGC. The algorithm also extends, with the loss of an extra $\log (1/ε)$ factor, to PCSPs on larger domains with a certain structural condition, which is implied by, e.g., a family of Plurality polymorphisms. We prove that assuming the UGC, robust satisfiability is preserved under the addition of equality constraints. As a consequence, we can extend the rich algebraic techniques for decision/search PCSPs to robust PCSPs. The methods involve the development of a correlated and robust version of the general SDP rounding algorithm for CSPs due to (Brown-Cohen, Raghavendra, ICALP 2016), which might be of independent interest.

翻译：本文延续了(Brakensiek, Guruswami, Sandeep, STOC 2023 / Discrete Analysis 2025)中开启的承诺约束满足问题(PCSPs)鲁棒可满足性研究，并取得以下成果：对于带否定的PCSP 1-in-3-SAT与NAE-SAT问题，我们在唯一博弈猜想(UGC)下证明，对于$(1-ε)$-可满足的实例，满足$1-Ω(1/\log (1/ε))$约束是困难的。这表明BGS算法在交替阈值多态情形下产生的指数级损失是不可避免的，而多数多态情形可实现多项式损失。对于任何允许多数多态的布尔PCSP，当承诺存在满足$1-ε$较强约束的赋值时，我们给出了满足$1-O(\sqrtε)$较弱约束的算法。这显著推广了Charikar--Makarychev--Makarychev针对2-SAT的算法，并匹配了UGC下可能的最优权衡。该算法还可推广至具有特定结构条件的大定义域PCSPs（例如由复数多态族隐含的条件），但需额外损失$\log (1/ε)$因子。我们证明在UGC假设下，鲁棒可满足性在添加等式约束后保持不变。由此可将决策/搜索PCSPs的丰富代数技术扩展至鲁棒PCSPs。方法的核心是发展了(Brown-Cohen, Raghavendra, ICALP 2016)提出的CSP通用SDP舍入算法的相关鲁棒版本，该成果可能具有独立研究价值。