A polyhedral approach to the multi-layer crossing minimization problem

Extended abstract
  • Michael Jünger
  • Eva K. Lee
  • Petra Mutzel
  • Thomas Odenthal
Planarity and Crossings
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1353)


We study the multi-layer crossing minimization problem from a polyhedral point of view. After the introduction of an integer programming formulation of the multi-layer crossing minimization problem, we examine the 2-layer case and derive several classes of facets of the associated polytope. Preliminary computational results for 2- and 3-layer instances indicate, that the usage of the corresponding facet-defining inequalities in a branch-and-cut approach may only lead to a practically useful algorithm, if deeper polyhedral studies are conducted.


Bipartite Graph Integer Program Short Path Problem Integer Programming Formulation Crossing Minimization 
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  1. Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Algorithms for drawing graphs: An annotated bibliography. Computational Geometry: Theory and Applications 4 (1994) 235–282.MathSciNetGoogle Scholar
  2. Eades, P., Kelly, D.: Heuristics for Reducing Crossings in 2-Layered Networks. Ars Combinatoria 21-A (1986) 89–98.Google Scholar
  3. Eades, P., Wormald, N.C.: Edge Crossings in Drawings of Bipartite Graphs. Algorithmica 11 (1994) 379–403.CrossRefGoogle Scholar
  4. Eades, P., McKay, B.D., Wormald, N.C.: On an edge crossing problem. Proc. 9th Australian Computer Science Conference, Australian National University (1986) 327–334.Google Scholar
  5. Fukuda, K.: Face Lattices. Personal Communication (1996).Google Scholar
  6. Garey, M.R., Johnson, D.S.: Crossing Number is NP-Complete. SIAM Journal on Algebraic and Discrete Methods 4 (1983) 312–316.Google Scholar
  7. Grötschel, M., Jünger, M., Reinelt, G.: Facets of the linear ordering polytope. Mathematical Programming 33 (1985) 43–60.CrossRefGoogle Scholar
  8. Harary, F.: Determinants, permanents and bipartite graphs. Mathematical Magazine 42 (1969) 146–148.Google Scholar
  9. Harary, F., Schwenk, A.: A new crossing number for bipartite graphs. Utilitas Mathematica 1 (1972) 203–209.Google Scholar
  10. Himsolt, M.: Personal Communication (1997).Google Scholar
  11. Jünger, M., Mutzel, P.: 2-Layer Straightline Crossing Minimization: Performance of Exact and Heuristic Algorithms. Journal of Graph Algorithms and Applications (JGAA), (, No. 1, Vol. 1, (1997) 125.Google Scholar
  12. Kusiak, A., Wang, J.: Dependency Analysis in Constraint Negotiation. IEEE Trans. Sys. Man, Cybern. 25 (1995) 1301–1313.Google Scholar
  13. May, M., Szkatula, K.: On the bipartite crossing number. Control and Cybernetics 17 No.1 (1988) 85–97.Google Scholar
  14. Mutzel, P.: An Alternative Method for Crossing Minimization. Lecture Notes in Computer Science LNCS 1190 (1997) 318–333.Google Scholar
  15. Richter, B.R., Thomassen, C.: A survey on crossing numbers. Manuscript, Carleton University and The Technical University of Denmark (1994).Google Scholar
  16. Shahrokhi, F., Szelky, L.A., Vrtô, L: Crossing Number of Graphs, Lower Bound Techniques and Algorithms: A Survey. Lecture Notes in Computer Science LNCS 894 (1995) 131–142.Google Scholar
  17. Shieh, F., McCreary, C.,: Directed Graphs Drawing by Clan-Based Decomposition. Lecture Notes in Computer Science LNCS 1027 (1996) 472–482.Google Scholar
  18. Sugiyama, K., Tagawa, S., Toda, M.: Methods for Visual Understanding of Hierarchical System Structures. IEEE Trans. Syst. Man, Cybern. SMC-11 (1981) 109–125.Google Scholar
  19. Tomii, N., Kambayashi, Y., Shunzo, Y.: On Planarization Algorithms of 2-Level Graphs. Papers of tech. group on electronic computers, IECEJ, EC77-38 (1977) 1–12.Google Scholar
  20. Valls, V., Marti, R., Lino, P.: A Branch and Bound Algorithm for Minimizing the Number of Crossing Arcs in Bipartite Graphs. Journal of Operational Research 90 (1996a) 303–319.CrossRefGoogle Scholar
  21. Valls, V., Marti, R., Lino, P.: A tabu thresholding algorithm for arc crossing minimization in bipartite graphs. Annals of Operations Research 83 (1996b) 223–251.Google Scholar
  22. Warfield, J.N.: Crossing Theory and Hierarchy Mapping. IEEE Trans. Syst. Man, Cybern. SMC-7 (1977) 505–523.Google Scholar
  23. Watkins, M.E.: A special crossing number for bipartite graphs: a research problem. Annals of New York Academy of Sciences 175 (1970) 405–410.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Michael Jünger
    • 1
  • Eva K. Lee
    • 2
  • Petra Mutzel
    • 3
  • Thomas Odenthal
    • 4
  1. 1.Institut für InformatikUniversität zu KölnGermany
  2. 2.Ind. & Sys. Eng.Georgia Institute of TechnologyUSA
  3. 3.Max-Planck-Institut für InformatikSaarbrücken
  4. 4.Ind. Eng. & Op. Res.Columbia UniversityUSA

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