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We discuss formulations of integer programs with a huge number of variables and their solution by column generation methods, i.e., implicit pricing of nonbasic variables to generate new columns or to prove LP optimality at a node of the branch-and-bound tree. We present classes of models for which this approach decomposes the problem, provides tighter LP relaxations, and eliminates symmetry. We then discuss computational issues and implementation of column generation, branch-and-bound algorithms, including special branching rules and efficient ways to solve the LP relaxation. We also discuss the relationship with Lagrangian duality.

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In the course of the deliberations of the 1977 Lanchester Prize Committee, Alan J. Goldman brought to our attention an error in the proof of Lemma 1 of our paper (Cornuejols, G., M. L. Fisher, G. L. Nemhauser. 1977. Location of bank accounts to optimize float: an analytic study of exact and approximate algorithms. Management Sci. 23 789–810.). The lemma, however, is true and the original correct, but long and intricate, proof was provided to the Committee, see (Cornuejols, G., M. L. Fisher, G. L. Nemhauser. 1977. On the uncapacitated location problem. Ann. Discrete Math. 1 163–178.) for details.