A Faster Exact Separation Algorithm for Blossom Inequalities

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In 1982, Padberg and Rao gave a polynomial-time separation algorithm for b-matching polyhedra. The current best known implementations of their separation algorithm run in \({\cal O}(|V|^2|E| \log (|V|^2/|E|))\) time when there are no edge capacities, but in \({\cal O}(|V||E|^2 \log (|V|^2/|E|))\) time when capacities are present.

We propose a new exact separation algorithm for the capacitated case which has the same running time as for the uncapacitated case. For the sake of brevity, however, we will restrict our introduction to the case of perfect 1-capacitated b-matchings, which includes, for example, the separation problem for perfect 2-matchings. As well as being faster than the Padberg-Rao approach, our new algorithm is simpler and easier to implement.