A database D may be inconsistent wrt a given set IC of integrity constraints. Consistent Query Answering is the problem of computing from D the answers to a query that are consistent wrt IC . Consistent answers have been characterized as those that are invariant under certain minimal forms of restoration of the consistency of the database. In this paper we investigate algorithmic and complexity theoretic issues of CQA under database repairs that minimally depart -wrt the cardinality of the symmetric difference- from the original database. Research on this kind of repairs had been suggested in the literature, but no systematic study had been done. Here we obtain first tight complexity bounds.