We consider the problem of satisfiability of sets of constraints in a given set of finite uniform hypergraphs. While the problem under consideration is similar in nature to the problem of satisfiability of constraints in graphs, the classical complexity reduction to finite-domain CSPs that was used in the proof of the complexity dichotomy for such problems cannot be used as a black box in our case. We therefore introduce an algorithmic technique inspired by classical notions from the theory of finite-domain CSPs, and prove its correctness based on symmetries that depend on a linear order that is external to the structures under consideration. Our second main result is a P/NP-complete complexity dichotomy for such problems over many sets of uniform hypergraphs. The proof is based on the translation of the problem into the framework of constraint satisfaction problems (CSPs) over infinite uniform hypergraphs. Our result confirms in particular the Bodirsky-Pinsker conjecture for CSPs of first-order reducts of some homogeneous hypergraphs. This forms a vast generalization of previous work by Bodirsky-Pinsker (STOC'11) and Bodirsky-Martin-Pinsker-Pongr\'acz (ICALP'16) on graph satisfiability.

翻译：考虑有限一致超图集合中的约束可满足性问题。尽管该问题在本质上与图中的约束可满足性类似，但用于此类问题复杂性二分性证明的经典归约方法——将问题归约至有限域CSP问题——在本研究中无法直接作为黑箱工具使用。为此，我们引入一种受有限域CSP理论经典概念启发的算法技术，并基于依赖于所考虑结构外部线性序的对称性，证明了该算法的正确性。我们的第二个主要成果是针对多类一致超图集合，建立了该问题的P/NP完全复杂性二分性定理。该证明通过将问题转化为无限一致超图上的约束满足问题（CSP）框架来实现。特别地，该结果印证了Bodirsky-Pinsker关于某些齐次超图一阶约化结构的CSP猜想。这构成了Bodirsky-Pinsker (STOC'11)与Bodirsky-Martin-Pinsker-Pongrácz (ICALP'16)关于图可满足性研究的广泛推广。