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 yet found in textbooks. We emphasize the growing understanding of the dual point of view, which has brought considerable progress to the column generation theory and practice. It stimulated careful initializations, sophisticated solution techniques for the restricted master problem and subproblem, as well as better overall performance. Thus, the dual perspective is an ever recurring concept in our "selected topics."

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Selected Topics in Column Generation

The vehicle routing problem

Entity resolution is central to data integration and data cleaning. Algorithmic approaches have been improving in quality, but remain far from perfect. Crowdsourcing platforms offer a more accurate but expensive (and slow) way to bring human insight into the process. Previous work has proposed batching verification tasks for presentation to human workers but even with batching, a human-only approach is infeasible for data sets of even moderate size, due to the large numbers of matches to be tested. Instead, we propose a hybrid human-machine approach in which machines are used to do an initial, coarse pass over all the data, and people are used to verify only the most likely matching pairs. We show that for such a hybrid system, generating the minimum number of verification tasks of a given size is NP-Hard, but we develop a novel two-tiered heuristic approach for creating batched tasks. We describe this method, and present the results of extensive experiments on real data sets using a popular crowdsourcing platform. The experiments show that our hybrid approach achieves both good efficiency and high accuracy compared to machine-only or human-only alternatives.

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CrowdER: crowdsourcing entity resolution

I The Early Years.- Solution of a Large-Scale Traveling-Salesman Problem.- The Hungarian Method for the Assignment Problem.- Integral Boundary Points of Convex Polyhedra.- Outline of an Algorithm for Integer Solutions to Linear Programs An Algorithm for the Mixed Integer Problem.- An Automatic Method for Solving Discrete Programming Problems.- Integer Programming: Methods, Uses, Computation.- Matroid Partition.- Reducibility Among Combinatorial Problems.- Lagrangian Relaxation for Integer Programming.- Disjunctive Programming.- II From the Beginnings to the State-of-the-Art.- Polyhedral Approaches to Mixed Integer Linear Programming.- Fifty-Plus Years of Combinatorial Integer Programming.- Reformulation and Decomposition of Integer Programs.- III Current Topics.- Integer Programming and Algorithmic Geometry of Numbers.- Nonlinear Integer Programming.- Mixed Integer Programming Computation.- Symmetry in Integer Linear Programming.- Semidefinite Relaxations for Integer Programming.- The Group-Theoretic Approach in Mixed Integer Programming.

50 years of integer programming 1958-2008 : from the early years to the state-of-the-art

We give a didactic introduction to the use of the column generation technique in linear and in particular in integer programming. We touch on both, the relevant basic theory and more advanced ideas which help in solving large scale practical problems. Our discussion includes embedding Dantzig-Wolfe decomposition and Lagrangian relaxation within a branch-and-bound framework, deriving natural branching and cutting rules by means of a so-called compact formulation, and understanding and influencing the behavior of the dual variables during column generation. Most concepts are illustrated via a small example. We close with a discussion of the classical cutting stock problem and some suggestions for further reading.

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A Primer in Column Generation

We explore an arc flow formulation with side constraints for the one‐dimensionalbin‐packing problem. The model has a set of flow conservation constraints and a set ofconstraints that force the appropriate number of items to be included in the packing. Themodel is tightened by fixing some variables at zero level, to reduce the symmetry of thesolution space, and by introducing valid inequalities. The model is solved exactly using abranch‐and‐price procedure that combines deferred variable generation and branch‐and‐bound.At each iteration, the subproblem generates a set of columns, which altogether correspondto an attractive valid packing for a single bin. We describe this subproblem, and theway it is modified in the branch‐and‐bound phase, after the branching constraints are addedto the model. We report the computational times obtained in the solution of the bin‐packingproblems from the OR‐Library test data sets. The linear relaxation of this model provides astrong lower bound for the bin‐packing problem and leads to tractable branch‐and‐boundtrees for the instances under consideration.

J. M. Valério de Carvalho

Papers

Exact solution of bin‐packing problems using column generation and branch‐and‐bound

LP models for bin packing and cutting stock problems

Solving the vehicle routing problem with time windows and multiple routes exactly using a pseudo-polynomial model

An integer programming model for two- and three-stage two-dimensional cutting stock problems

Arc-flow model for the two-dimensional guillotine cutting stock problem