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International Symposium on Experimental Algorithms

SEA 2016: Experimental Algorithms pp 75–88Cite as

A Practical Method for the Minimum Genus of a Graph: Models and Experiments

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Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9685))

Abstract

We consider the problem of the minimum genus of a graph, a fundamental measure of non-planarity. We propose the first formulations of this problem as an integer linear program (ILP) and as a satisfiability problem (SAT). These allow us to develop the first working implementations of general algorithms for the problem, other than exhaustive search. We investigate several different ways to speed-up and strengthen the formulations; our experimental evaluation shows that our approach performs well on small to medium-sized graphs with small genus, and compares favorably to other approaches.

M. Chimani—Supported by the German Research Foundation (DFG) project CH 897/2-1.

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Notes

  1. 1.

    For a simple graph, the minimum genus embedding contains no face of length 1 or 2. On the other hand, we cannot be more specific than the lower bound of 3.

  2. 2.

    In [10], the validity of such a preprocessing is shown for several non-planarity measures, namely crossing number, skewness, coarseness, and thickness. Let H be the NPC of G. We can trivially observe that (A) \(\gamma (G)\le \gamma (H)\), and (B) \(\gamma (G)\ge \gamma (H)\). A: Given an optimal solution for H, we can embed each S onto the surface in place of its replacement edge, without any crossings. B: Each replaced component S contains a path connecting its poles that is drawn crossing-free in the optimal embedding of G; we can planarly draw all of S along this path, and then simplify the embedding by replacing this locally drawn S by its replacement edge; this gives a solution for H on the same surface.

  3. 3.

    First term: each edge lies on at most two faces, each face has size at least 3; second term: Euler’s formula with genus at least 1.

  4. 4.

    The previous version was the winner of the Sequential Appl. SAT+UNSAT Track of the SAT competition 2014 [3]. This improved version is even faster.

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Acknowledgements

We thank Armin Biere for providing the most recent version (as of 2015-06-05) of the lingeling SAT solver.

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Authors and Affiliations

  1. Institute of Computer Science, University of Osnabrück, Osnabrück, Germany

    Stephan Beyer, Markus Chimani & Ivo Hedtke

  2. Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark

    Michal Kotrbčík

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  1. Stephan Beyer
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  2. Markus Chimani
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Correspondence to Stephan Beyer or Ivo Hedtke .

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  1. Amazon.com, Inc., Palo Alto, California, USA

    Andrew V. Goldberg

  2. Russian Academy of Sciences, St. Petersburg, Russia

    Alexander S. Kulikov

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Beyer, S., Chimani, M., Hedtke, I., Kotrbčík, M. (2016). A Practical Method for the Minimum Genus of a Graph: Models and Experiments. In: Goldberg, A., Kulikov, A. (eds) Experimental Algorithms. SEA 2016. Lecture Notes in Computer Science(), vol 9685. Springer, Cham. https://doi.org/10.1007/978-3-319-38851-9_6

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  • DOI: https://doi.org/10.1007/978-3-319-38851-9_6

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