Polynomial Algorithms—Matroids

Matroid theory has generated an array of useful results and insights that are central to the field of combinatorial optimization. Like many other elegant mathematical structures, matroids admit a notion of duality. The matroid concept of duality subsumes many dualities for special cases. A planar graph is a graph that can be drawn in the plane without edge crossings. Such a drawing is called the plane imbedding of the graph. The computation needed to check independence varies greatly with the nature of the underlying matroid. The independence system, defined by the edge set of an undirected graph and the collection of cycle-free subsets of the edges, is a matroid. This chapter presents an algorithm for the problem of determining a maximum cardinality intersection of two matroids. This algorithmic approach is a primal-dual one. There are essentially four key elements that provide a summary of the primal-dual scheme.