In this first chapter we motivate our method via the assignment problem. Through this problem, we highlight the basic ingredients and ideas of the method. We then give an outline of how a typical chapter in the rest of the book is structured, and how the remaining chapters are organized.

The assignment problem

Consider the classical assignment problem: Given a bipartite graph G = (V1 ∪ V2, E) with |V1| = |V2| and weight function w: E → ℝ+, the objective is to match every vertex in V1 with a distinct vertex in V2 to minimize the total weight (cost) of the matching. This is also called the minimum weight bipartite perfect matching problem in the literature and is a fundamental problem in combinatorial optimization. See Figure 1.1 for an example of a perfect matching in a bipartite graph.

One approach to the assignment problem is to model it as a linear programming problem. A linear program is a mathematical formulation of the problem with a system of linear constraints that can contain both equalities and inequalities, and also a linear objective function that is to be maximized or minimized. In the assignment problem, we associate a variable xuv for every {u, v} ∈ E. Ideally, we would like the variables to take one of two values, zero or one (hence in the ideal case, they are binary variables). When xuv is set to one, we intend the model to signal that this pair is matched; when xuv is set to zero, we intend the model to signal that this pair is not matched.