  Operational consistent query answering (CQA) is a recent framework for CQA
based on revised definitions of repairs, which are built by applying a sequence
of operations (e.g., fact deletions) starting from an inconsistent database
until we reach a database that is consistent w.r.t. the given set of
constraints. It has been recently shown that there are efficient approximations
for computing the percentage of repairs, as well as of sequences of operations
leading to repairs, that entail a given query when we focus on primary keys,
conjunctive queries, and assuming the query is fixed (i.e., in data
complexity). However, it has been left open whether such approximations exist
when the query is part of the input (i.e., in combined complexity). We show
that this is the case when we focus on self-join-free conjunctive queries of
bounded generelized hypertreewidth. We also show that it is unlikely that
efficient approximation schemes exist once we give up one of the adopted
syntactic restrictions, i.e., self-join-freeness or bounding the generelized
hypertreewidth. Towards the desired approximation schemes, we introduce a novel
counting complexity class, called SpanTL, show that each problem in SpanTL
admits an efficient approximation scheme by using a recent approximability
result in the context of tree automata, and then place the problems of interest
in SpanTL.
