Total Dual Integrality of Matching Forest Constraints

G

=(V, E, A) be a mixed graph. That is, (V, E) is an undirected graph and (V, A) is a directed graph. A matching forest (introduced by R. Giles) is a subset F of such that F contains no circuit (in the underlying undirected graph) and such that for each there is at most one such that v is head of e. (For an undirected edge e, both ends of e are called head of e.) Giles gave a polynomial-time algorithm to find a maximum-weight matching forest, yielding as a by-product a characterization of the inequalities determining the convex hull of the incidence vectors of the matching forests. We prove that these inequalities form a totally dual integral system. It is equivalent to an ``all-integer'' min-max relation for the maximum weight of a matching forest. Our proof is based on an exchange property for matching forests, and implies Giles' characterization.