The Dynamic Complexity of Acyclic Hypergraph Homomorphisms

Finding a homomorphism from some hypergraph $\mathcal{Q}$ (or some relational structure) to another hypergraph $\mathcal{D}$ is a fundamental problem in computer science. We show that an answer to this problem can be maintained under single-edge changes of $\mathcal{Q}$, as long as it stays acyclic, in the DynFO framework of Patnaik and Immerman that uses updates expressed in first-order logic. If additionally also changes of $\mathcal{D}$ are allowed, we show that it is unlikely that existence of homomorphisms can be maintained in DynFO.