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## Branch-and-price: Column generation for solving huge integer programs (1998)

Venue: | OPER. RES |

Citations: | 347 - 13 self |

### Citations

1376 | Integer and Combinatorial Optimization”, - Nemhauser, Wolsey - 1999 |

376 |
Decomposition principle for linear programs
- Dantzig, Wolfe
- 1960
(Show Context)
Citation Context ...points of its convex hull, denoted by conv(S ). Since representing a bounded polyhedron by its extreme points is the basic construct of Dantzig-Wolfe decomposition and generalized linear programming [=-=Dantzig and Wolfe, 1960-=-], our column generation approach to integer programming is closely related to Dantzig-Wolfe decomposition and the earlier work on path flows in networks of Ford and Fulkerson [1958]. When S is unboun... |

213 | Optimization Theory for Large Systems, - Lasdon - 1970 |

186 | Lagrangean relaxation for integer programming. - Geoffrion - 1974 |

180 | A new optimization algorithm for the vehicle routing problem with time windows, - Desrochers, Desrosiers, et al. - 1992 |

168 | The Travelling-Salesman Problem and Minimum Spanning Trees: - Held, Karp - 1971 |

162 | H.P.: Validation of subgradient optimization - Held, Wolfe, et al. - 1974 |

161 | Time constrained routing and scheduling, - Desrosiers, Dumas, et al. - 1995 |

130 |
On the facial structure of set packing polyhedra.
- Padberg
- 1973
(Show Context)
Citation Context ... the generalized assignment problem. The restricted master problem is a set partitioning problem and a well-known class of valid inequalities for the set partitioning problem are clique inequalities [=-=Padberg 1973-=-], which simply say in linear programming terms that the sum of the variables in the clique cannot exceed one. Now consider the pricing problem after the addition of a clique inequality to the restric... |

129 | The traveling salesman problem - Junger, Reinelt, et al. - 1997 |

103 | Set partitioning: a survey - Balas, Padberg - 1976 |

89 | An Integer programming approach to scheduling. - Ryan, Foster - 1981 |

86 | The pickup and delivery problem with time windows - Dumas, Desrosiers, et al. - 1991 |

78 | A Branch-And-Cut Algorithm for the Generalized Assignment Problem, - Savelsbergh - 1997 |

61 | A suggested computation for the maximal multicommodity network flows - Ford, Fulkerson - 1958 |

53 | Solving binary cutting stock problems by column generation and branch-and-bound - Vance, Barnhart, et al. - 1994 |

49 | A column generation approach to the urban transit crew scheduling problem - Desrochers, Soumis - 1989 |

46 | Airline Crew Scheduling: A New Formulation and Decomposition Algorithm - Vance, Barnhart, et al. - 1997 |

45 | Routing with time windows by column generation, - Desrosiers, Soumis, et al. - 1984 |

42 | Very large-scale linear programming: A case study in combining interior point and simplex method - Bixby, Gregory, et al. - 1992 |

39 | LP-Based Combinatorial Problem Solving - Hoffman, Padberg - 1985 |

32 | A column generation algorithm for a ship scheduling problem - Appelgren - 1969 |

30 | Functional Description of MINTO, a Mixed INTeger Optimizer, Version 2.0b.” Report COC-91-03 from Computational Optimization
- Savelsbergh, Nemhauser
- 1994
(Show Context)
Citation Context ...e solving an LP but also allow the embedding of column generation LP solving into a general branch-and-bound structure for solving MIPs. The use of MINTO [Nemhauser, Savelsbergh, and Sigismondi 1994, =-=Savelsbergh and Nemhauser 1993-=-] may reduce the implementation e orts even further. MINTO (Mixed INTeger Optimizer) is based on the belief that to solve large mixed-integer programs e ciently, without having to develop a full-blown... |

27 | A Global Approach to Crew Pairing Optimization. - Anbil, Tanga, et al. - 1992 |

26 | Nondifferentiable Optimization - Lemaréchal - 1989 |

26 | Decomposition and Column Generation for Integer Programs - Vanderbeck - 1994 |

19 | A column generation algorithm for bandwidth packing - Parker, Ryan - 1994 |

19 | Solution Techniques for LongHaul Crew Assignment Problem - Bamhart, Johnson, et al. - 1992 |

19 | G.: Min-cut clustering - Johnson, Mehrotra, et al. - 1993 |

19 | The cutting stock problem and integer rounding - Marcotte - 1985 |

18 | A note on bounding a class of linear programming problems, including cutting stock problems - Farley - 1990 |

18 |
Modeling and strong linear programs for mixed integer programming
- Johnson
- 1989
(Show Context)
Citation Context ...969] observed 25 years ago, it is not that straightforward. There are fundamental difficulties in applying column generation techniques for linear programming in integer programming solution methods [=-=Johnson 1989-=-]. These include: ffl Conventional integer programming branching on variables may not be effective because fixing variables can destroy the structure of the pricing problem. ffl Solving these LPs to o... |

16 |
Crew scheduling, cutting stock and column generation: Solving huge integer programs
- Vance
- 1993
(Show Context)
Citation Context ... form pairings. The major di culty is the total number of pairings, which grows exponentially with the number of ights. For example, in a typical problem with 253 ights, there are 5,833,004 pairings [=-=Vance 1993-=-]. However, it is possible to represent pairings as paths in a graph, and to evaluate their costs with a multilabel shortest path or dynamic programming algorithm, see Desrochers and Soumis [1989], Ba... |

15 | M.: Totally balanced and greedy matrices - Hoffman, Kolen, et al. - 1985 |

14 | A time-indexed formulation for single-machine scheduling problems: Column generation - Akker, Hurkens, et al. - 2000 |

13 | A polyhedral approach to edge coloring - Nemhauser, Park - 1991 |

10 | Slyke - Dantzig, Van - 1967 |

8 | An alternate formulation and solution strategy for multi-commodity network flow problems - BARNHART, JOHNSON, et al. - 1991 |

8 | Primal Partition Programming for Block Diagonal Matrices - Rosen - 1964 |

8 | Nondi®erentiable optimization - Lemar¶echal - 1989 |

7 | Finding Everett’s Lagrange multipliers by linear programming - Brooks, Geoffrion - 1966 |

6 | An improved dual-based algorithm for the generalized assignment problem - Guignard, Rosenwein - 1989 |

5 | Constrained Graph Partitioning: Decomposition, Polyhedral Structure and Algorithms - Mehrotra - 1992 |

5 |
A branch-and-price algorithm for the pickup and delivery problem with time windows
- Sol, Savelsbergh
(Show Context)
Citation Context ...ake existing columns with reduced cost equal to zero (at least all basic columns satisfy this requirement) and try to construct columns with a positive reduced cost by performing some simple changes [=-=Sol and Savelsbergh 1994-=-]. Notice the similaritybetween the column management functions performed in branchand-price algorithms and the row management functions performed in branch-and-cut algorithms. 5.3 Alternative bounds ... |

3 | Capacitated network design with column generation - Clarke, Gong - 1995 |

2 |
Crew Planning at Delta Airlines, presentation at
- Marsten
- 1994
(Show Context)
Citation Context ... giving a better representation of the face. Although no extensive computational tests have been done to investigate the differences, it seems that using interior point methods works somewhat better [=-=Marsten 1994-=-]. 23 6.5 LP solution The computationally most intensive component of a branch-and-price algorithm is the solution of the linear programs, which includes the solution of many pricing problems. Therefo... |

2 | On integer programming decomposition and ways to enforce integrality in the master - VANDERBECK - 1996 |

2 | An integer programming approach toscheduling - Ryan, Foster - 1981 |

1 | A Column Generation Approach to Graph Coloring. Graduate - Mehrotra, Trick - 1993 |

1 | A Column Generation ApproachtoGraph Coloring. Graduate - Mehrotra, Trick - 1993 |

1 | An optimal column generation with ranking algorithm for very large set partitioning problems in tra c assignment - Ribeiro, Minoux, et al. - 1989 |