  This paper deals with faces and facets of the family-variable polytope and
the characteristic-imset polytope, which are special polytopes used in integer
linear programming approaches to statistically learn Bayesian network
structure. A common form of linear objectives to be maximized in this area
leads to the concept of score equivalence (SE), both for linear objectives and
for faces of the family-variable polytope. We characterize the linear space of
SE objectives and establish a one-to-one correspondence between SE faces of the
family-variable polytope, the faces of the characteristic-imset polytope, and
standardized supermodular functions. The characterization of SE facets in terms
of extremality of the corresponding supermodular function gives an elegant
method to verify whether an inequality is SE-facet-defining for the
family-variable polytope. We also show that when maximizing an SE objective one
can eliminate linear constraints of the family-variable polytope that
correspond to non-SE facets. However, we show that solely considering SE facets
is not enough as a counter-example shows; one has to consider the linear
inequality constraints that correspond to facets of the characteristic-imset
polytope despite the fact that they may not define facets in the
family-variable mode.
