Using ILP/SAT to Determine Pathwidth, Visibility Representations, and other Grid-Based Graph Drawings

  • Therese Biedl
  • Thomas Bläsius
  • Benjamin Niedermann
  • Martin Nöllenburg
  • Roman Prutkin
  • Ignaz Rutter
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8242)


We present a simple and versatile formulation of grid-based graph representation problems as an integer linear program (ILP) and a corresponding SAT instance. In a grid-based representation vertices and edges correspond to axis-parallel boxes on an underlying integer grid; boxes can be further constrained in their shapes and interactions by additional problem-specific constraints. We describe a general d-dimensional model for grid representation problems. This model can be used to solve a variety of NP-hard graph problems, including pathwidth, bandwidth, optimum st-orientation, area-minimal (bar-k) visibility representation, boxicity-k graphs and others. We implemented SAT-models for all of the above problems and evaluated them on the Rome graphs collection. The experiments show that our model successfully solves NP-hard problems within few minutes on small to medium-size Rome graphs.


  1. 1.
  2. 2.
    Biedl, T., Bläsius, T., Niedermann, B., Nöllenburg, M., Prutkin, R., Rutter, I.: A versatile ILP/SAT formulation for pathwidth, optimum st-orientation, visibility representation, and other grid-based graph drawing problems. CoRR abs/1308.6778 (2013)Google Scholar
  3. 3.
    Biere, A., Heule, M., van Maaren, H., Walsh, T. (eds.): Handbook of Satisfiability. IOS Press (2009)Google Scholar
  4. 4.
    Binucci, C., Didimo, W., Liotta, G., Nonato, M.: Orthogonal drawings of graphs with vertex and edge labels. Comput. Geom. Theory Appl. 32(2), 71–114 (2005)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Bodlaender, H.L., Gilbert, J.R., Hafsteinsson, H., Kloks, T.: Approximating treewidth, pathwidth, frontsize, and shortest elimination tree. J. Algorithms 18(2), 238–255 (1995)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Bodlaender, H.L., Kloks, T.: Efficient and constructive algorithms for the pathwidth and treewidth of graphs. J. Algorithms 21(2), 358–402 (1996)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Brandes, U.: Eager st-ordering. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 247–256. Springer, Heidelberg (2002)CrossRefGoogle Scholar
  8. 8.
    Buchheim, C., Chimani, M., Ebner, D., Gutwenger, C., Jünger, M., Klau, G.W., Mutzel, P., Weiskircher, R.: A branch-and-cut approach to the crossing number problem. Discrete Optimization 5(2), 373–388 (2008)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Chen, D.S., Batson, R.G., Dang, Y.: Applied Integer Programming. Wiley (2010)Google Scholar
  10. 10.
    Chimani, M., Mutzel, P., Bomze, I.: A new approach to exact crossing minimization. In: Halperin, D., Mehlhorn, K. (eds.) ESA 2008. LNCS, vol. 5193, pp. 284–296. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  11. 11.
    Chimani, M., Zeranski, R.: Upward planarity testing via SAT. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 248–259. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  12. 12.
    Chinn, P.Z., Chvátalova, J., Dewdney, A.K., Gibbs, N.E.: The bandwidth problem for graphs and matrices—a survey. J. Graph Theory 6(3), 223–254 (1982)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Dean, A.M., Evans, W., Gethner, E., Laison, J.D., Safari, M.A., Trotter, W.T.: Bar k-visibility graphs. J. Graph Algorithms Appl. 11(1), 45–59 (2007)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Del Corso, G.M., Manzini, G.: Finding exact solutions to the bandwidth minimization problem. Computing 62(3), 189–203 (1999)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Dujmovic, V., Fellows, M.R., Kitching, M., Liotta, G., McCartin, C., Nishimura, N., Ragde, P., Rosamond, F.A., Whitesides, S., Wood, D.R.: On the parameterized complexity of layered graph drawing. Algorithmica 52(2), 267–292 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Eén, N., Sörensson, N.: An extensible SAT-solver. In: Giunchiglia, E., Tacchella, A. (eds.) SAT 2003. LNCS, vol. 2919, pp. 502–518. Springer, Heidelberg (2004)CrossRefGoogle Scholar
  17. 17.
    Even, S., Tarjan, R.E.: Computing an st-numbering. Theoret. Comput. Sci. 2(3), 339–344 (1976)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Felsner, S., Liotta, G., Wismath, S.: Straight-line drawings on restricted integer grids in two and three dimensions. J. Graph Algorithms Appl. 7(4), 363–398 (2003)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Gange, G., Stuckey, P.J., Marriott, K.: Optimal k-level planarization and crossing minimization. In: Brandes, U., Cornelsen, S. (eds.) GD 2010. LNCS, vol. 6502, pp. 238–249. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  20. 20.
    Gurobi Optimization, Inc.: Gurobi optimizer reference manual (2013)Google Scholar
  21. 21.
    Hopcroft, J., Tarjan, R.: Efficient planarity testing. J. ACM 21(4), 549–568 (1974)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Jünger, M., Mutzel, P.: 2-layer straightline crossing minimization: Performance of exact and heuristic algorithms. J. Graph Algorithms Appl. 1(1), 1–25 (1997)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kratochvíl, J.: A special planar satisfiability problem and a consequence of its NP-completeness. Discrete Appl. Math. 52(3), 233–252 (1994)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Lin, X., Eades, P.: Towards area requirements for drawing hierarchically planar graphs. Theoret. Comput. Sci. 292(3), 679–695 (2003)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Martí, R., Campos, V., Piñana, E.: A branch and bound algorithm for the matrix bandwidth minimization. Europ. J. of Operational Research 186, 513–528 (2008)CrossRefMATHGoogle Scholar
  26. 26.
    Nöllenburg, M., Wolff, A.: Drawing and labeling high-quality metro maps by mixed-integer programming. IEEE TVCG 17(5), 626–641 (2011)Google Scholar
  27. 27.
    Papamanthou, C., G. Tollis, I.: Applications of parameterized st-orientations. J. Graph Algorithms Appl. 14(2), 337–365 (2010)MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Sadasivam, S., Zhang, H.: NP-completeness of st-orientations for plane graphs. Theoret. Comput. Sci. 411(7-9), 995–1003 (2010)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Tamassia, R., Tollis, I.: A unified approach to visibility representations of planar graphs. Discrete Comput. Geom. 1(1), 321–341 (1986)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Wismath, S.K.: Characterizing bar line-of-sight graphs. In: Proc. First Ann. Symp. Comput. Geom., SCG 1985, pp. 147–152. ACM, New York (1985)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  • Therese Biedl
    • 1
  • Thomas Bläsius
    • 2
  • Benjamin Niedermann
    • 2
  • Martin Nöllenburg
    • 2
  • Roman Prutkin
    • 2
  • Ignaz Rutter
    • 2
  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooCanada
  2. 2.Institute of Theoretical InformaticsKarlsruhe Institute of TechnologyGermany

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