An alternative method to crossing minimization on hierarchical graphs

Extended abstract
  • Petra Mutzel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1190)


A common method for drawing directed graphs is, as a first step, to partition the vertices into a set of k levels and then, as a second step, to permute the vertices within the levels such that the number of crossings is minimized. We suggest an alternative method for the second step, namely, removing the minimal number of edges such that the resulting graph is k-level planar. For the final diagram the removed edges are reinserted into a k-level planar drawing. Hence, instead of considering the k-level crossing minimization problem, we suggest solving the k-level planarization problem. In this paper we address the case k=2. First, we give a motivation for our approach. Then, we address the problem of extracting a 2-level planar subgraph of maximum weight in a given 2-level graph. This problem is NP-hard. Based on a characterization of 2-level planar graphs, we give an integer linear programming formulation for the 2-level planarization problem. Moreover, we define and investigate the polytope \(2\mathcal{L}\mathcal{P}\mathcal{S}\)(G) associated with the set of all 2-level planar subgraphs of a given 2-level graph G. We will see that this polytope has full dimension and that the inequalities occuring in the integer linear description are facet-defining for \(2\mathcal{L}\mathcal{P}\mathcal{S}\)(G). The inequalities in the integer linear programming formulation can be separated in polynomial time, hence they can be used efficiently in a cutting plane method for solving practical instances of the 2-level planarization problem. Furthermore, we derive new inequalities that substantially improve the quality of the obtained solution. We report on first computational results.


  1. [BN88]
    G. Di Battista and E. Nardelli: Hierarchies and Planarity Theory. IEEE Transactions on Systems, Man and Cybernetics 18 (1988) 1035–1046Google Scholar
  2. [Car80]
    Carpano, M.J.: Automatic display of hierarchized graphs for computer aided decision analysis. IEEE Transactions on Systems, Man and Cybernetics, SMC-10, no. 11 (1980) 705–715Google Scholar
  3. [Edm70]
    Edmonds, J.: Submodular functions, matroids and certain polyhedra. in: Combinatorial Structures and Their Applications, Gordon and Breach, London (1970) 69–87Google Scholar
  4. [Dre94]
    Dresbach, S.: A New Heuristic Layout Algorithm for DAGs. in: Derigs, Bachem & Drexl (eds.), Operations Research Proceedings 1994, Springer Verlag, Berlin (1994) 121–126Google Scholar
  5. [EK86]
    Eades, P., and D. Kelly: Heuristics for Reducing Crossings in 2-Layered Networks. Ars Combinatoria 21-A (1986) 89–98Google Scholar
  6. [EKW86]
    Eades, P., B.D. McKay, and N.C. Wormald: On an edge crossing problem. Proc. 9th Australian Computer Science Conference, Australian National University (1986) 327–334Google Scholar
  7. [EW94a]
    Eades, P., and N.C. Wormald: Edge crossings in Drawings of Bipartite Graphs. Algorithmica 10 (1994) 379–403CrossRefGoogle Scholar
  8. [EW94b]
    Eades, P. and S. Whitesides: Drawing graphs in two layers. Theoretical Computer Science 131 (1994) 361–374CrossRefGoogle Scholar
  9. [Fuk96]
    Fukuda, A.: Face Lattices. Personal Communication (1996)Google Scholar
  10. [GLS81]
    Grötschel, M., L. Lovász, and A. Schrijver: The Ellipsoid Method and its Consequences in Combinatorial Optimization. Combinatorica 1 (1981) 169–197Google Scholar
  11. [GP85]
    Grötschel, M. and M.W. Padberg: Polyhedral theory. In E.L. Lawler, J.K. Lenstra, A.H.G. Rinnoy Kan, and D.B. Shmoys (eds.), The Traveling Salesman Problem: A Guided Tour of Combinatorial Optimization, Wiley-Interscience (1985)Google Scholar
  12. [HP96]
    Heath, L.S. and S.V. Pemmaraju: Recognizing Leveled-Planar Dags in Linear Time. Lecture Notes in Comp. Sci. 1027, in: F. Brandenburg (ed.), Proceedings on Graph Drawing '95, Passau (1996) 300–311Google Scholar
  13. [JM93a]
    Jünger, M. und P. Mutzel: Solving the maximum planar subgraph problem by branch and cut. In: L.A. Wolsey and G. Rinaldi (eds.), Proceedings of the 3rd IPCO Conference, Erice (1993) 479–492Google Scholar
  14. [JM93b]
    Jünger, M. und P. Mutzel: Maximum planar subgraphs and nice embeddings: Practical layout tools. Algorithmica 16, No. 1, Special Issue on Graph Drawing, G. Di Battista and R. Tamassia (eds.), (1996) 33–59, also Report No. 93.145, Universität zu Köln, (1993)Google Scholar
  15. [JM96]
    Jünger, M. und P. Mutzel: Exact and Heuristic Algorithms for 2-Layer Straightline Crossing Minimization. Lecture Notes in Comp. Sci. 1027, in: F. Brandenburg (ed.), Proc. on Graph Drawing '95, Passau (1996) 337–348Google Scholar
  16. [JRT95]
    Jünger, M., G. Reinelt, and S. Thienel: Practical Problem Solving with Cutting Plane Algorithms in Combinatorial Optimization. DIMACS Series in Discrete Mathematics and Theoretical Computer Science, 20 (1995) 111–152Google Scholar
  17. [KP80]
    Karp, R.M. and C.H. Papadimitriou: On Linear Characterizations of Combinatorial Optimization Problems. Proc. of the 21st Annual Symp. on the Foundations of Computer Science IEEE (1980) 1–9Google Scholar
  18. [Len90]
    Lengauer, T.: Combinatorial algorithms for integrated circuit layout. John Wiley & Sons, Chichester, UK (1990)Google Scholar
  19. [Mäk90]
    Mäkinen, E.: Experiments on Drawing 2-Level Hierarchical Graphs. Intern. J. Computer Math. 37 (1990) 129–135Google Scholar
  20. [Mut94]
    Mutzel, P.: The Maximum Planar Subgraph Problem. Dissertation, Universität zu Köln (1994)Google Scholar
  21. [PR81]
    Padberg, M.W. and M.R. Rao: The Russian Method for Linear Inequalities III: Bounded Integer Programming. GBA Working Paper 81–39, New York University (1981)Google Scholar
  22. [PW83]
    Padberg, M.W. and L.A. Wolsey: Trees and Cuts. Annals of Discrete Mathematics 17 (1983) 511–517Google Scholar
  23. [STT81]
    Sugiyama, K., S. Tagawa, and M. Toda: Methods for Visual Understanding of Hierarchical System Structures. IEEE Trans. Syst. Man, Cybern., SMC-11 (1981) 109–125Google Scholar
  24. [TKY77]
    Tomii, N., Y. Kambayashi, and Y. Shuzo: On Planalization Algorithms of 2-Level Graphs. Papers of tech. group on electronic computers, IECEJ, EC77-38 (1977) 1–12Google Scholar
  25. [U1184]
    Ullman, J.D.: Computational Aspects of VLSI. Computer Science Press, Rockville, MD (1984)Google Scholar
  26. [VLM96]
    Vingron, M., H.-P. Lenhof, and P. Mutzel: Computational Molecular Biology. In: Annotated Bibliographies in Combinatorial Optimization, M. Dell'Amico, F. Maffioli, S. Martello (eds.), Chapter 23, to appear (1996)Google Scholar
  27. [VML96]
    Valls, V., R. Marti, and P. Lino: A Branch and Bound Algorithm for Minimizing the Number of Crossing Arcs in Bipartite Graphs. Journal of Operational Research 90 (1996) 303–319CrossRefGoogle Scholar
  28. [War77]
    Warfield, J.N.: Crossing Theory and Hierarchy Mapping. IEEE Trans. Syst. Man, Cybern., SMC-7 (1977) 505–523Google Scholar
  29. [WG86]
    Waterman, M.S. and J. R. Griggs: Interval graphs and maps of DNA. Bull. Math. Biology 48, no. 2 (1986) 189–195Google Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Petra Mutzel
    • 1
  1. 1.Max-Planck-Institut für InformatikSaarbrückenGermany

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