  We present a class of linear programming approximations for constrained
optimization problems. In the case of mixed-integer polynomial optimization
problems, if the intersection graph of the constraints has bounded tree-width
our construction yields a class of linear size formulations that attain any
desired tolerance. As a result, we obtain an approximation scheme for the
"AC-OPF" problem on graphs with bounded tree-width. We also describe a more
general construction for pure binary optimization problems where individual
constraints are available through a membership oracle; if the intersection
graph for the constraints has bounded tree-width our construction is of linear
size and exact. This improves on a number of results in the literature, both
from the perspective of formulation size and generality.
