Chapter 3 Matching

Abstract

Publisher Summary Matching is pairing: dividing a collection of objects into pairs. Typically the objective is to maximize total profit (or minimize cost), where the profit of each possible pair is known in advance. A matching problem is defined and classified by two parameters: the graph G and the weights w. Matching theory is one of the cornerstones of mathematical programming. The chapter discusses examples of matching problems. The two obvious directions in which the matching problem can be generalized are: degree-constrained optimization and set-packing problems. The chapter considers algorithms for finding maximum cardinality as well as maximum weight matchings. It start with the Hungarian method for finding a maximum cardinality matching in a bipartite graph. The method is extended in two directions: to Edmonds' blossom algorithm for finding a maximum cardinality matching in a general graph and to the Hungarian method for finding a maximum weight matching in a bipartite graph. Finally, the ideas of these two methods are combined in Edmonds' algorithm for the weighted matching problem in general graphs. The chapter considers general degree constraints and discusses some of the self-refining aspects of matching. Other algorithms for matching problems, including randomized algorithms for maximum matching and for counting matching is presented. The chapter concludes with discussion on the computer implementation of matching algorithms and on heuristics for matching problems.