A value of a CSP instance is typically defined as a fraction of constraints that can be simultaneously met. We propose an alternative definition of a value of an instance and show that, for purely combinatorial reasons, a value of an unsolvable instance is bounded away from one; we call this fact a gap theorem. We show that the gap theorem implies NP-hardness of a gap version of the Layered Label Cover Problem. The same result can be derived from the PCP Theorem, but a full, self-contained proof of our reduction is quite short and the result can still provide PCP-free NP-hardness proofs for numerous problems. The simplicity of our reasoning also suggests that weaker versions of Unique-Games-type conjectures, e.g., the d-to-1 conjecture, might be accessible and serve as an intermediate step for proving these conjectures in their full strength. As the second, main application we provide a sufficient condition under which a fixed template Promise Constraint Satisfaction Problem (PCSP) reduces to another PCSP. The correctness of the reduction hinges on the gap theorem, but the reduction itself is very simple. As a consequence, we obtain that every CSP can be canonically reduced to most of the known NP-hard PCSPs, such as the approximate hypergraph coloring problem.
翻译:CSP 实例的值通常被定义为可以同时满足的制约值的一小部分。 我们建议了对实例值的替代定义, 并表明, 出于纯粹的组合性原因, 一个无法解决实例的值与一个参数相隔开; 我们称这个事实为空标。 我们显示, 差距理论意味着多层标签覆盖问题的空白版本的NP- 硬性。 同样的结果也可以来自五氯苯酚理论, 但一个完整的、 自成一体的减排证据非常短, 结果仍然可以为很多问题提供无五氯苯酚的NP- 硬性证明。 我们推理的简单性也表明, 弱化的Unique- Games 类型配置的版本, 例如, d- to-1 符号, 可能是可以利用的, 并且作为中间步骤, 证明这些组合的完整。 第二, 我们的主要应用提供了一个充分的条件, 固定的模板“ 承诺满意度问题( PCSP) 能够向另一个 PCSP PCSP 提供无五氯苯酚的无硬性证明。 我们的精确性解释性能将一个简单化的C- 缩小到最深层次。