## Packing and Partitioning Orbitopes

Please always quote using this URN: urn:nbn:de:0297-zib-9104

- We introduce orbitopes as the convex hulls of 0/1-matrices that are lexicographically maximal subject to a group acting on the columns. Special cases are packing and partitioning orbitopes, which arise from restrictions to matrices with at most or exactly one 1-entry in each row, respectively. The goal of investigating these polytopes is to gain insight into ways of breaking certain symmetries in integer programs by adding constraints, e.g., for a well-known formulation of the graph coloring problem. We provide a thorough polyhedral investigation of packing and partitioning orbitopes for the cases in which the group acting on the columns is the cyclic group or the symmetric group. Our main results are complete linear inequality descriptions of these polytopes by facet-defining inequalities. For the cyclic group case, the descriptions turn out to be totally unimodular, while for the symmetric group case, both the description and the proof are more involved. The associated separation problems can be solved in linear time.

Author: | Volker Kaibel, Marc PfetschORCiD |
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Document Type: | ZIB-Report |

Tag: | integer programming; lexicographic representatives; symmetry breaking |

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: | 2006/03/28 |

Series (Serial Number): | ZIB-Report (06-17) |

ZIB-Reportnumber: | 06-17 |

Published in: | Appeared in: Mathematical Programming 114 (2008) 1-36 |