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

An order out of nowhere: a new algorithm for infinite-domain CSPs