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

We consider constraint satisfaction problems whose relations are defined in first-order logic over any uniform hypergraph satisfying certain weak abstract structural conditions. Our main result is a P/NP-complete complexity dichotomy for such CSPs. Surprisingly, the large class of structures under consideration falls into a mixed regime where neither the classical complexity reduction to finite-domain CSPs can be used as a black box, nor does the class exhibit order properties, known to prevent the application of this reduction. We 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.