## Orbitopal Fixing

Please always quote using this URN: urn:nbn:de:0297-zib-9422

- The topic of this paper are integer programming models in which a subset of 0/1-variables encode a partitioning of a set of objects into disjoint subsets. Such models can be surprisingly hard to solve by branch-and-cut algorithms if the permutation of the subsets of the partition is irrelevant. This kind of symmetry unnecessarily blows up the branch-and-cut tree. We present a general tool, called orbitopal fixing, for enhancing the capabilities of branch-and-cut algorithms in solving this kind of symmetric integer programming models. We devise a linear time algorithm that, applied at each node of the branch-and-cut tree, removes redundant parts of the tree produced by the above mentioned permutations. The method relies on certain polyhedra, called orbitopes, which have been investigated in (Kaibel and Pfetsch (2006)). However, it does not add inequalities to the model, and thus, it does not increase the difficulty of solving the linear programming relaxations. We demonstrate the computational power of orbitopal fixing at the example of a graph partitioning problem motivated from frequency planning in mobile telecommunication networks.

Author: | Volker Kaibel, Matthias Peinhardt, Marc E. Pfetsch |
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Document Type: | ZIB-Report |

Tag: | orbitopes; symmetry breaking; variable fixing |

MSC-Classification: | 52-XX CONVEX AND DISCRETE GEOMETRY / 52Bxx Polytopes and polyhedra / 52B12 Special polytopes (linear programming, centrally symmetric, etc.) |

90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C10 Integer programming | |

90-XX OPERATIONS RESEARCH, MATHEMATICAL PROGRAMMING / 90Cxx Mathematical programming [See also 49Mxx, 65Kxx] / 90C57 Polyhedral combinatorics, branch-and-bound, branch-and-cut | |

Date of first Publication: | 2006/11/17 |

Series (Serial Number): | ZIB-Report (06-48) |

Published in: | Appeared in: Proc. of the 12th Integer Programming and Combinatorial Optimization Conference (IPCO) M. Fischetti and D. Williamson (eds.), LNCS 4513, Springer-Verlag, 74-88 |