Mathematical Programming

, Volume 175, Issue 1–2, pp 197–240 | Cite as

Polytopes associated with symmetry handling

  • Christopher HojnyEmail author
  • Marc E. Pfetsch
Full Length Paper Series A


This paper investigates a polyhedral approach to handle symmetries in mixed-binary programs. We study symretopes, i.e., the convex hulls of all binary vectors that are lexicographically maximal in their orbit with respect to the symmetry group. These polytopes turn out to be quite complex. For practical use, we therefore develop an integer programming formulation with ternary coefficients, based on knapsack polytopes arising from a single lexicographic order enforcing inequality. We show that for these polytopes, the optimization as well as the separation problem of minimal cover inequalities can be solved in almost linear time. We demonstrate the usefulness of this approach by computational experiments, showing that it is competitive with state-of-the-art methods and is considerably faster for specific problem classes.

Mathematics Subject Classification

90C09 90C11 90C57 



We thank Andreas Paffenholz for helpful discussions. Moreover, we thank two anonymous referees for their valuable suggestions that helped to improve this paper. In particular, we thank one referee for an idea to improve the running time of the algorithm in Theorem 23.

Supplementary material


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Research Group OptimizationTU DarmstadtDarmstadtGermany

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