  Making cut generating functions (CGFs) computationally viable is a central
question in modern integer programming research. One would like to find CGFs
that are simultaneously good, i.e., there are good guarantees for the cutting
planes they generate, and efficient, meaning that the values of the CGFs can be
computed cheaply (with procedures that have some hope of being implemented in
current solvers). We investigate in this paper to what extent this balance can
be struck. We propose a family of CGFs which, in a sense, achieves this harmony
between good and efficient. In particular, we provide a parameterized family of
$b+\Z^n$ free sets to derive CGFs from and show that our proposed CGFs give a
good approximation of the closure given by CGFs obtained from all maximal
$b+\Z^n$ free sets and their so-called trivial liftings, and simultaneously,
show that these CGFs can be computed with explicit, efficient procedures. We
provide a constructive framework to identify these sets as well as computing
their trivial lifting. We follow it up with computational experiments to
demonstrate this and to evaluate their practical use. Our proposed family of
cuts seem to give some tangible improvement on randomly generated instances
compared to GMI cuts; however, in MIPLIB 3.0 instances, and vertex cover and
stable problems on random graph instances, their performance is poor.
