Matchings

Abstract

GTC has five managers Anna, Boris, Caren, Derek, and Elija, labeled A, B, C, D, and E, respectively. It also has five projects labeled P1 through P5. Figure 4.1 shows the ability of each of the executives to handle projects – a link between a manager and a project indicates that the manager has the skill set required to handle the project. A manager can handle at most one project, and each project, if assigned to a manager, needs to be assigned to exactly one manager. GTC wants to find out how to assign projects to managers such that the maximum number of projects will be assigned.

This problem, and others similar to it in nature in which solutions correspond to pairing of entities are referred to as matching problems. The solutions to these problems are called matchings. Formally stated, matchings are subsets of edges in a network such that no two edges in the set are incident on the same node of the network. Thus, a solution to GTC’s problem would be the set of edges {A – P₃, B – P₂, C – P₅}.