In this work, we revisit binary polynomial optimization (BPO) problems with
limited treewidth of the associated graph. We provide alternate proofs of the
exactness of a reformulated linear program (LP) with O(n.2^d) variables, n
being the number of variables and d being the treewidth of the associated
graph. The first proof relies on expressing any given fractional solution as the
midpoint of two points in the feasible region in order to prove integrality. The
second proof is a primal-dual complementary slackness argument explicitly
constructing optimal solutions to primal and dual variables.