Algorithmic characterization of bipartite b-matching and matroid intersection

An algorithmic characterization of a particular combinatorial optimization problem means that there is an algorithm that works exact if and only if applied to the combinatorial optimization problem under investigation. According to Jack Edmonds, the Greedy algorithm leads to an algorithmic characterization of matroids. We deal here with the algorithmic characterization of the intersection of two matroids. To this end we introduce two different augmentation digraphs for the intersection of any two independence systems. Paths and cycles in these digraphs correspond to candidates for improving feasible solutions. The first digraph gives rise to an algorithmic characterization of bipartite b-matching. The second digraph leads to a polynomial-time augmentation algorithm for the (weighted) matroid intersection problem and to a conjecture about an algorithmic characterization of matroid intersection.