Parallel Computation of Combinatorial Symmetries

Authors Markus Anders, Pascal Schweitzer

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Markus Anders
  • TU Darmstadt, Germany
Pascal Schweitzer
  • TU Darmstadt, Germany


We thank Adolfo Piperno, Brendan McKay, Tommi Junttila, and Petteri Kaski for discussions providing us with deeper insights into their isomorphism solvers. We also want to thank our colleagues Thomas Schneider, Jendrik Brachter, and Moritz Lichter for the fruitful discussions we had on some of the topics in this paper.

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Markus Anders and Pascal Schweitzer. Parallel Computation of Combinatorial Symmetries. In 29th Annual European Symposium on Algorithms (ESA 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 204, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In practice symmetries of combinatorial structures are computed by transforming the structure into an annotated graph whose automorphisms correspond exactly to the desired symmetries. An automorphism solver is then employed to compute the automorphism group of the constructed graph. Such solvers have been developed for over 50 years, and highly efficient sequential, single core tools are available. However no competitive parallel tools are available for the task. We introduce a new parallel randomized algorithm that is based on a modification of the individualization-refinement paradigm used by sequential solvers. The use of randomization crucially enables parallelization. We report extensive benchmark results that show that our solver is competitive to state-of-the-art solvers on a single thread, while scaling remarkably well with the use of more threads. This results in order-of-magnitude improvements on many graph classes over state-of-the-art solvers. In fact, our tool is the first parallel graph automorphism tool that outperforms current sequential tools.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Shared memory algorithms
  • graph isomorphism
  • automorphism groups
  • algorithm engineering
  • parallel algorithms


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