  In the context of ontology-mediated querying with description logics (DLs),
we study the data complexity of queries in which selected predicates can be
closed (OMQCs). We provide a non-uniform analysis, aiming at a classification
of the complexity into tractable and non-tractable for ontologies in the
lightweight DLs DL-Lite and EL, and the expressive DL ALCHI. At the level of
ontologies, we prove a dichotomy between FO-rewritable and coNP-complete for
DL-Lite and between PTime and coNP-complete for EL. The meta problem of
deciding tractability is proved to be in PTime. At the level of OMQCs, we show
that there is no dichotomy (unless NP equals PTime) if both concept and role
names can be closed. If only concept names can be closed, we tightly link the
complexity of query evaluation to the complexity of surjective CSPs. We also
identify a class of OMQCs based on ontologies formulated in DL-Lite that are
guaranteed to be tractable and even FO-rewritable.
