Exact and heuristic algorithms for 2-layer straightline crossing minimization

  • Michael Jünger
  • Petra Mutzel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1027)


We present algorithms for the two layer straightline crossing minimization problem that are able to compute exact optima. Our computational results lead us to the conclusion that there is no need for heuristics if one layer is fixed, even though the problem is NP-hard, and that for the general problem with two variable layers, true optima can be computed for sparse instances in which the smaller layer contains up to 15 nodes. For bigger instances, the iterated barycenter method turns out to be the method of choice among several popular heuristics whose performance we could assess by comparing the results to optimum solutions.


  1. [CPLEX]
    CPLEX: Using the CPLEX callable library and the CPLEX mixed integer library. CPLEX Optimization Inc. (1993)Google Scholar
  2. [D94]
    Dresbach, S.: A New Heuristic Layout Algorithm for DAGs. Derigs, Bachern & Drexl (eds.) Operations Research Proceedings 1994, Springer Verlag, Berlin (1994) 121–126Google Scholar
  3. [D95]
    Dresbach, S.: Personal communication. (1995)Google Scholar
  4. [EK86]
    Eades, P., and D. Kelly: Heuristics for Reducing Crossings in 2-Layered Networks. Ars Combinatoria 21-A (1986) 89–98Google Scholar
  5. [EW94]
    Eades, P., and N.C. Wormald: Edge crossings in Drawings of Bipartite Graphs. Algorithmica 10 (1994) 379–403CrossRefGoogle Scholar
  6. [GJ83]
    Garey, M.R., and D.S. Johnson: Crossing Number is NP-Complete. SIAM J. on Algebraic and Discrete Methods 4 (1983) 312–316Google Scholar
  7. [GJR84a]
    Grötschel, M., M. Jünger, and G. Reinelt: A cutting plane algorithm for the linear ordering problem. Operations Research 32 (1984) 1195–1220Google Scholar
  8. [GJR84b]
    Grötschel, M., M. Jünger, and G. Reinelt: Optimal triangulation of large real world input-output matrices. Statistische Hefte 25 (1984) 261–295Google Scholar
  9. [GJR85]
    Grötschel, M., M. Jünger, and G. Reinelt: Facets of the linear ordering polytope. Mathematical Programming 33 (1985) 43–60CrossRefGoogle Scholar
  10. [K93]
    Knuth, D.E.: The Stanford GraphBase: A Platform for Combinatorial Computing. ACM Press, Addison-Wesley Publishing Company (1993) New YorkGoogle Scholar
  11. [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
  12. [W77]
    Warfield, J.N.: Crossing Theory and Hierarchy Mapping. IEEE Trans. Syst. Man, Cybern., SMC-7 (1977) 505–523Google Scholar

Copyright information

© Springer-Verlag 1996

Authors and Affiliations

  • Michael Jünger
    • 1
  • Petra Mutzel
    • 2
  1. 1.Institut für InformatikUniversität zu KölnKölnGermany
  2. 2.Max-Planck-Institut für InformatikSaarbrückenGermany

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