Odd minimum cut sets and b-matchings revisited

The famous Padberg-Rao separation algorithm for $b$-matching polyhedra can be implemented to run in $\mathcal{O}(|V|^2|E|\log(|V|^2/|E|))$ time in the uncapacitated case, and in $\mathcal{O}(|V||E|^2\log(|V|^2/|E|))$ time in the capacitated case. We give a new and simple algorithm for the capacitated case which can be implemented to run in $\mathcal{O}(|V|^2|E|\log(|V|^2/|E|))$ time.