Mathematical Programming Computation

, Volume 11, Issue 1, pp 37–93 | Cite as

A computational comparison of symmetry handling methods for mixed integer programs

  • Marc E. PfetschEmail author
  • Thomas Rehn
Full Length Paper


The handling of symmetries in mixed integer programs in order to speed up the solution process of branch-and-cut solvers has recently received significant attention, both in theory and practice. This paper compares different methods for handling symmetries using a common implementation framework. We start by investigating the computation of symmetries and analyze the symmetries present in the MIPLIB 2010 instances. It turns out that many instances are affected by symmetry and most symmetry groups contain full symmetric groups as factors. We then present (variants of) six symmetry handling methods from the literature. Their implementation is tested on several testsets. On very symmetric instances used previously in the literature, it is essential to use methods like isomorphism pruning, orbital fixing, or orbital branching. Moreover, tests on the MIPLIB instances show that isomorphism pruning, orbital fixing, or adding symmetry breaking inequalities allow to speed-up the solution process by about 15% and more instances can be solved within the time limit.


Symmetry Mixed integer program Branch-and-cut Isomorphism pruning Orbital branching 

Mathematics Subject Classification

90C11 90C57 



We thank Tobias Achterberg for interesting discussions on the topic and Christopher Hojny for helpful comments. We also thank the editor and referees for their helpful comments that helped to improve this paper. Furthermore, the first author acknowledges support of the German Research Foundation (DFG) within the Collaborative Research Center 666.

Supplementary material

12532_2018_140_MOESM1_ESM.pdf (854 kb)
Supplementary material 1 (pdf 853 KB)


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Copyright information

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

Authors and Affiliations

  1. 1.Fachbereich MathematikTechnische Universität DarmstadtDarmstadtGermany
  2. 2.initOS GmbHMagdeburgGermany

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