Dantzig-Wolfe decomposition and column generation, devised for linear programs, is a success story in large scale integer programming. We outline and relate the approaches, and survey mainly recent contributions, not found in textbooks, yet. We emphasize on the growing understanding of the dual point of view, which brought considerable progress to the column generation theory and practice. It stimulated careful initializations, sophisticated solution techniques for restricted master problem and subproblem, as well as better overall performance. Thus, the dual perspective is an ever recurring concept in our "selected topics."

During the eigthies and early nineties, the best exact algorithms for the Capacitated Vehicle Routing Problem (CVRP) utilized lower bounds obtained by Lagrangean relaxation or column generation. Next, the advances in the polyhedral description of the CVRP yielded branch-andcut algorithms giving better results. However, several instances in the range of 50–80 vertices, some proposed more than 30 years ago, can not be solved with current known techniques. This paper presents an algorithm utilizing a lower bound obtained by minimizing over the intersection of the polytopes associated to a traditional Lagrangean relaxation over q-routes and the one defined by bounds, degree and the capacity constraints. This is equivalent to a linear program with an exponential number of both variables and constraints. Computational experiments show the new lower bound to be superior to the previous ones, specially when the number of vehicles is large. The resulting branch-and-cut-and-price could solve to optimality almost all instances from the literature up to 100 vertices, nearly doubling the size of the instances that can be consistently solved. Further progress in this algorithm may be soon obtained by also using other known families of inequalities. 1

The Generalized Assignment Problem (GAP) is a classic scheduling problem with many applications. We propose a branch-and-cut-andprice for that problem featuring a stabilization mechanism to accelerate column generation convergence. We also propose ellipsoidal cuts, a new way of transforming the exact algorithm into a powerful heuristic, in the same spirit of the cuts recently proposed by Fischetti and Lodi. The improved solutions found by this heuristic can, in turn, help the task of the exact algorithm. The resulting algorithms showed a very good performance and were able to solve three among the last five open instances from the OR-Library. 1

Decomposition algorithms such as Lagrangian relaxation and Dantzig-Wolfe decomposition are well-known methods that can be used to generate bounds for mixed-integer linear programming problems. Traditionally, these methods have been viewed as distinct from polyhedral methods, in which bounds are obtained by dynamically generating valid inequalities to strengthen an initial linear programming relaxation. Recently, a number of authors have proposed methods for integrating dynamic cut generation with various decomposition methods to yield further improvement in computed bounds. In this paper, we describe a framework within which most of these methods can be viewed from a common theoretical perspective. We then discuss how the framework can be extended to obtain a decomposition-based separation technique we call decompose and cut. As a by-product, we describe how these methods can take advantage of the fact that solutions with known structure, such as those to a given relaxation, can frequently be separated much more easily than arbitrary real vectors. 1

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