Branch and Price: Integer Programming with Column Generation

Branch-and-price is a generalization of linear programming (LP) based branch-and-bound specifically designed to handle integer programming (IP) formulations that contain a huge number of variables. The basic idea of branch-and-price is simple. Columns are left out of the LP relaxation because there are too many columns to handle efficiently and most of them will have their associated variable equal to zero in an optimal solution anyway. Then to check the optimality of an LP solution, a subproblem, called the pricing problem, is solved to try to identify columns with a profitable reduced cost. If such columns are found, the LP is reoptimized. Branching occurs when no profitable columns are found, but the LP solution does not satisfy the integrality conditions. Branch-and-price applies column generation at every node of the branch-and-bound tree. There are several reasons for considering IP formulations with a huge number of variables. • A compact formulation of an IP may have a weak LP relaxation. Frequently the relaxation can be tightened by a reformula-tion that involves a huge number of variables. • A compact formulation of an IP may have a symmetric structure that causes branch-and-bound to perform poorly because the problem barely changes after branching. A reformulation with a huge number of variables can eliminate this symmetry. • Column generation provides a decomposition of the problem into master and sub-problems. This decomposition may have a natural interpretation in the contextual setting allowing for the incorporation of additional important constraints or nonlin-ear cost functions. • A formulation with a huge number of variables may be the only choice. At first glance, it may seem that branch-and-price involves nothing more than combining well-known ideas for solving linear programs by column generation with branch-and-bound. However, it is not that straightforward. There are fundamental difficulties in applying column generation techniques for linear programming in integer programming solution methods. These include: • Conventional integer programming branching on variables may not be effective because fixing variables can destroy the structure of the pricing problem. • Column generation often converges slowly and solving the LPs to optimality may become computationally prohibitive. We illustrate the concepts of branch-and-price and the difficulties that may arise by means of an example. In the generalized assignment problem (GAP) the objective is to find a maximum profit assignment of m tasks to n machines such that each task is assigned to precisely one machine subject to capacity restrictions …