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## Conflict Graphs in Integer Programming (1998)

Venue: | EUROPEAN JOURNAL OF OPERATIONS RESEARCH |

Citations: | 10 - 2 self |

### Citations

1997 |
Reducibility among combinatorial problems
- Karp
- 1972
(Show Context)
Citation Context ...B + r a rk \Gamma maxfwz : z vertex packing of G(U)g: 4 The summation term P k2B + r a rk is added to the bound since we use the complements for B + r . The weighted vertex packing problem is NP-hard =-=[8]-=-. Therefore, in order to get a lower bound on L r , we solve a relaxation of the vertex packing problem. The relaxation is based on the observation that if a graph consists of a set of disjoint comple... |

152 |
Solving airline crew scheduling problems by branch-and-cut,"
- Hoffman, Padberg
- 1993
(Show Context)
Citation Context ...ing techniques can be implemented to run much faster for constraints with 0-1 coefficients only. Furthermore, pure 0-1 rows allow additional preprocessing techniques, such as row domination, see e.g. =-=[1, 5, 7]-=-. 5 Cut generation Any feasible solution to S defines a vertex packing in the conflict graph. Therefore, the vertex packing polytope associated with the conflict graph contains the convex hull of feas... |

131 |
On the facial structure of set packing polyhedra".
- Padberg
- 1973
(Show Context)
Citation Context ...(P )) is the stability number of the subgraph induced by P . We limit ourselves to clique inequalities, i.e., inequalities with ff(G(P )) = 1. Clique inequalities have been shown to be facet defining =-=[10]-=- for the vertex packing polytope. The separation problem for the class of clique inequalities is max P aeV f\Sigma j2P w j z j : ff(G(P )) = 1g and is a maximum weighted clique problem on the conflict... |

79 | An updated mixed integer programming library: MIPLIB 3.0
- Bixby, Ceria, et al.
- 1998
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Citation Context ...one on an IBM RS/6000 Model 590 workstation with one hour CPU time limit. Our data set consists of problems with varying characteristics. The first five problems are various instances from MIPLIB 3.0 =-=[2]-=-. The next five are real-life production planning and resource allocation problems. The next set of five problems are instances of a timeindexed formulation of a single machine scheduling problem. Due... |

69 |
Preprocessing and probing techniques for mixed integer programming problems.
- Savelsbergh
- 1994
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Citation Context ...s, we explain these techniques in more detail. 2.1 Implications from feasibility conditions Probing refers to setting a binary variable to one of its bounds tentatively and examining the consequences =-=[3, 11]-=-. When we tentatively set a binary variable to one of its bounds 3 and a subsequent analysis shows that the problem has become infeasible, then we can fix that variable to its opposite bound. Formally... |

48 |
a Mixed INTeger Optimizer
- MINTO
- 1994
(Show Context)
Citation Context ...nts conducted to test the effectiveness and efficiency of the implementation techniques and algorithms described in the previous sections. Our implementation is embedded in MINTO (version 3.0). MINTO =-=[9]-=- is a software system that solves mixed-integer linear programs by a branch-and-bound algorithm with linear programming relaxations. All experiments were done on an IBM RS/6000 Model 590 workstation w... |

44 |
Improving LP-representations of zero-one linear programs for branch-and-cut.
- Hoffman, Padberg
- 1991
(Show Context)
Citation Context ...ounds on variables, and to improve coefficients of constraints. Preprocessing may reduce the size of an instance as well as the integrality gap. Several papers have appeared on this subject; see e.g. =-=[4, 11] for -=-the specifics of these reformulation techniques. At the heart of these techniques is the computation of lower and upper bounds on the value of the left-hand side of constraints. For "less than or... |

22 |
Logical reduction methods in zero-one programming (minimal preferred variables
- Guignard, Spielberg
- 1981
(Show Context)
Citation Context ...s, we explain these techniques in more detail. 2.1 Implications from feasibility conditions Probing refers to setting a binary variable to one of its bounds tentatively and examining the consequences =-=[3, 11]-=-. When we tentatively set a binary variable to one of its bounds 3 and a subsequent analysis shows that the problem has become infeasible, then we can fix that variable to its opposite bound. Formally... |

21 | A combined lagrangian, linear programming, and implication heuristic for largescale set partitioning problems
- Atamturk, Nemhauser, et al.
- 1996
(Show Context)
Citation Context ...ing techniques can be implemented to run much faster for constraints with 0-1 coefficients only. Furthermore, pure 0-1 rows allow additional preprocessing techniques, such as row domination, see e.g. =-=[1, 5, 7]-=-. 5 Cut generation Any feasible solution to S defines a vertex packing in the conflict graph. Therefore, the vertex packing polytope associated with the conflict graph contains the convex hull of feas... |

19 |
Modelling and strong linear programs for mixed integer programming.
- Johnson
- 1989
(Show Context)
Citation Context ...ing techniques can be implemented to run much faster for constraints with 0-1 coefficients only. Furthermore, pure 0-1 rows allow additional preprocessing techniques, such as row domination, see e.g. =-=[1, 5, 7]-=-. 5 Cut generation Any feasible solution to S defines a vertex packing in the conflict graph. Therefore, the vertex packing polytope associated with the conflict graph contains the convex hull of feas... |

10 |
An improved branch-and-bound method for integer programming.
- Tomlin
- 1971
(Show Context)
Citation Context ...q p ! 0, x p = 0. So we obtain, \Delta = max q2Q min p2Pq (jc q j maxf1; j f qsa q p jg): (2) Hence, if z +sc i t i +sc j t j \Gamma \Delta ! z h , we have an edge inequality valid for S opt . Tomlin =-=[12]-=- used this line of argument in calculating penalties for choosing a branching variable. Since 7 both x i and x j are nonbasic, an edge inequality found in this way is not violated by the LP solution. ... |