Packing and partitioning orbitopes

By Volker Kaibel, Marc E. Pfetsch. 2006

140 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Industrial and Manufacturing Engineering

View Paper: https://www.semanticscholar.org/paper/816efb691113879d62c7361ea8c719b2e7784f15

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Citations in the corpus

The SCIP Optimization Suite 7.0 (2020, methodology): “The latter approach is implemented using three constraint handler plugins that take care of different kinds of symretopes: orbitopes [48], orbisacks [47, 56], and symresacks [41, 42].”
Symmetry in Integer Linear Programming (2010, methodology): “The partitioning orbitope, introduced in [61], is the polytope obtained as the convex hull of these rep-”