Promise Constraint Satisfaction Problems (PCSPs) are a generalization of Constraint Satisfaction Problems (CSPs) where each predicate has a strong and a weak form and given a CSP instance, the objective is to distinguish if the strong form can be satisfied vs. even the weak form cannot be satisfied. Since their formal introduction by Austrin, Guruswami, and H\aa stad, there has been a flurry of works on PCSPs [BBKO19,KO19,WZ20]. The key tool in studying PCSPs is the algebraic framework developed in the context of CSPs where the closure properties of the satisfying solutions known as the polymorphisms are analyzed. The polymorphisms of PCSPs are much richer than CSPs. In the Boolean case, we still do not know if dichotomy for PCSPs exists analogous to Schaefer's dichotomy result for CSPs. In this paper, we study a special case of Boolean PCSPs, namely Boolean Ordered PCSPs where the Boolean PCSPs have the predicate $x \leq y$. In the algebraic framework, this is the special case of Boolean PCSPs when the polymorphisms are monotone functions. We prove that Boolean Ordered PCSPs exhibit a computational dichotomy assuming the Rich 2-to-1 Conjecture [BKM21] which is a perfect completeness surrogate of the Unique Games Conjecture. Assuming the Rich 2-to-1 Conjecture, we prove that a Boolean Ordered PCSP can be solved in polynomial time if for every $\epsilon>0$, it has polymorphisms where each coordinate has Shapley value at most $\epsilon$, else it is NP-hard. The algorithmic part of our dichotomy is based on a structural lemma that Boolean monotone functions with each coordinate having low Shapley value have arbitrarily large threshold functions as minors. The hardness part proceeds by showing that the Shapley value is consistent under a uniformly random 2-to-1 minor. Of independent interest, we show that the Shapley value can be inconsistent under an adversarial 2-to-1 minor.

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布尔有序承诺CSP的条件二分法

承诺约束满足问题（PCSP）是约束满足问题（CSP）的泛化，其中每个谓词都有强形式和弱形式，并且在给定CSP实例的情况下，目标是区分是否可以满足强形式甚至弱形式。表格无法满足。自从Austrin，Guruswami和H \ aa stad正式介绍以来，已经有大量关于PCSP的作品[BBKO19，KO19，WZ20]。研究PCSP的关键工具是在CSP的背景下开发的代数框架，其中分析了令人满意的解的闭合性，即多态性。PCSP的多态性比CSP丰富得多。在布尔情况下，我们仍然不知道PCSP的二分法是否类似于Schaefer对CSP的二分法。在本文中，我们研究了布尔PCSP的一种特殊情况，即布尔有序PCSP，其中布尔PCSP具有谓词$ x \ leq y $。在代数框架中，当多态性是单调函数时，这是布尔型PCSP的特殊情况。我们证明了布尔顺序PCSP在假设Rich 2：1猜想[BKM21]的情况下表现出计算上的二分法，这是Unique Games猜想的完美完整性替代。假设Rich 2对1猜想，我们证明如果对于每个$ \ epsilon> 0 $，它具有多态性，其中每个坐标的Shapley值最多为$ \ epsilon $，那么布尔阶PCSP可以在多项式时间内求解。这是NP难的。我们二分法的算法部分基于结构引理，布尔单调函数的每个坐标的Shapley值都较低，而次要函数则具有任意大的阈值函数。硬度部分通过显示在均匀随机的2比1小调下Shapley值一致而继续进行。具有独立利益，我们表明在对抗性2对1小调下，Shapley值可能不一致。