  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.
