Lectures 4 and 5 – Matchings

A matching in a graph is a set of edges, no two of which share an endpoint. We describe Edmond’s algorithm for finding matchings of maximum cardinality in a graph. For bipartite graphs with edge weights, we also show how to find matchings of maximum weight. It is recommended that the reader draws pictures of the various constructs that we use (such as alternating trees). I was too lazy to incorporate pictures in these lecture notes. We observe that bipartite matchings are intersections of two (partition) matroids. Some algorithms presented here can serve as examples for the more general case of optimizing over intersections of two matroids. Finding the maximum intersection of three matroids includes the problem of 3-dimensional matching as a special case, and hence is NP-hard. Some algorithmic principles that we shall use are alternating paths and primal-dual algorithms.