A branch-and-cut approach to the crossing number problem

Discrete Optimization

Volumn 5, Issue 2, 2008, Pages 373-388

  Buchheim, Christoph ^a   Chimani, Markus ^b   Ebner, Dietmar ^c   Gutwenger, Carsten ^b   Jünger, Michael ^a   Klau, Gunnar W ^d,e   Mutzel, Petra ^b   Weiskircher, René ^f  

^a UNIVERSITY OF COLOGNE   (Germany)

^b TU DORTMUND UNIVERSITY   (Germany)

^c VIENNA UNIVERSITY OF TECHNOLOGY   (Austria)

^d FREIE UNIVERSITÄT BERLIN   (Germany)

^e Deutsche Forschungsgemeinschaft   (Germany)

^f CSIRO   (Australia)

Author keywords

Branch and cut; Column generation; Crossing number

Indexed keywords

COMPUTATIONAL METHODS; GRAPH THEORY; LEARNING ALGORITHMS; LINEAR PROGRAMMING; PROBLEM SOLVING;

BRANCH AND CUT APPROACH; CROSSING NUMBER; GRAPH CLASSES; PROGRAMMING FORMULATION;

NUMBER THEORY;

EID: 41149159803     PISSN: 15725286     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.disopt.2007.05.006     Document Type: Article

References (46)

1. Batini C., Talamo M., and Tamassia R. Computer aided layout of entity-relationship diagrams. Journal of Systems and Software 4 (1984) 163-173
2. Bhatt S.N., and Leighton F.T. A framework for solving VLSI graph layout problems. Journal of Computer and System Sciences 28 (1984) 300-343
3. H. Bodlaender, A. Grigoriev, Algorithms for graphs embeddable with few crossings per edge. Research Memoranda 036, Maastricht: METEOR, Maastricht Research School of Economics of Technology and Organization, 2004
4. Booth K.S., and Lueker G.S. Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. Journal of Computer and System Sciences 13 3 (1976) 335-379
5. Boyer J.M., and Myrvold W.J. On the cutting edge: Simplified O (n) planarity by edge addition. Journal of Graph Algorithms and Applications 8 3 (2004) 241-273
6. Cai J., Han X., and Tarjan R.E. An O (m log n)-time algorithm for the maximal planar subgraph problem. SIAM Journal on Computing 22 6 (1993) 1142-1162
7. Cǎlinescu G., Fernandes C.G., Finkler U., and Karloff H. A better approximation algorithm for finding planar subgraphs. Journal of Algorithms 27 2 (1998) 269-302
8. Codato1 G., and Fischetti M. Combinatorial benders cuts. In: Bienstock D., and Nemhauser G. (Eds). Integer Programming and Combinatorial Optimization IPCO 2004. Lecture Notes in Computer Science vol. 3064 (2004), Springer-Verlag 178-195
9. Dantzig G.B., and Wolfe P. Decomposition principle for linear programs. Operations Research 8 (1960) 101-111
10. de Fraysseix H., and de Mendez P.O. On cotree-critical and dfs cotree-critical graphs. Journal of Graph Algorithms and Applications 7 4 (2003) 411-427
11. Di Battista G., Garg A., Liotta G., Tamassia R., Tassinari E., and Vargiu F. An experimental comparison of four graph drawing algorithms. Computational Geometry: Theory and Applications 7 5-6 (1997) 303-325
12. Di Battista G., and Tamassia R. On-line maintenance of triconnected components with SPQR-trees. Algorithmica 15 (1996) 302-318
13. Di Battista G., and Tamassia R. On-line planarity testing. SIAM Journal on Computing 25 (1996) 956-997
14. Djidjev H.N. A linear algorithm for the maximal planar subgraph problem. Proceedings of the 4th Workshop on Algorithms and Data Structures. Lecture Notes in Computer Science vol. 955 (1995), Springer-Verlag 369-380
15. Eades P., and Wormald N.C. Edge crossings in drawings of bipartite graphs. Algorithmica 11 4 (1994) 379-403
16. Even S., and Tarjan R.E. Computing an st-numbering. Theoretical Computer Science 2 3 (1976) 339-344
17. Garey M.R., and Johnson D.S. Computers and Intractability: A Guide to the Theory of NP-Completeness (1979), W.H. Freeman, New York, NY
18. Garey M.R., and Johnson D.S. Crossing number is NP-complete. SIAM Journal on Algebraic and Discrete Methods 4 (1983) 312-316
19. M. Grohe, Computing crossing numbers in quadratic time, in: STOC 2001: Proceedings of the 33rd Annual ACM Symposium on Theory of Computing, 2001
20. Gutwenger C., and Chimani M. Non-planar core reduction of graphs. In: Eades P., and Healy P. (Eds). Graph Drawing 2005. Lecture Notes in Computer Science vol. 3843 (2006), Springer-Verlag 223-234
21. Gutwenger C., and Mutzel P. A linear time implementation of SPQR trees. In: Marks J. (Ed). Graph Drawing 2000. Lecture Notes in Computer Science vol. 1984 (2001), Springer-Verlag 77-90
22. Gutwenger C., and Mutzel P. An experimental study of crossing minimization heuristics. In: Liotta G. (Ed). Graph Drawing 2003. Lecture Notes in Computer Science vol. 2912 (2004), Springer-Verlag 13-24
23. Gutwenger C., Mutzel P., and Weiskircher R. Inserting an edge into a planar graph. Algorithmica 41 4 (2005) 289-308
24. Guy R.K. Crossing numbers of graphs. Graph Theory and Applications (Proceedings). Lecture Notes in Mathematics (1972), Springer-Verlag 111-124
25. Hooker J.N., Ottoson G., Thorsteinsson E.S., and Kim H.J. On integrating constraint propagation and linear programming for combinatorial optimization. Sixteenth National Conference on Artificial Intelligence AAAI-99 (1999), MIT Press 136-141
26. Hopcroft J.E., and Tarjan R.E. Dividing a graph into triconnected components. SIAM Journal on Computing 2 3 (1973) 135-158
27. Hopcroft J.E., and Tarjan R.E. Efficient planarity testing. Journal of the ACM 21 4 (1974) 549-568
28. W.-L. Hsu, A linear time algorithm for finding a maximal planar subgraph based on PC-trees, in: COCOON: Annual International Conference on Computing and Combinatorics, 2005, pp. 787-797
29. 2) algorithms for graph planarization. IEEE Transactions on Computer-Aided Design 8 (1989) 257-267
30. Jünger M., Leipert S., and Mutzel P. A note on computing a maximal planar subgraph using PQ-trees. IEEE Transactions on Computer-Aided Design of Integrated Circuits and Systems 17 7 (1998) 609-612
31. Jünger M., and Mutzel P. Maximum planar subgraphs and nice embeddings: Practical layout tools. Algorithmica 16 1 (1996) 33-59
32. Jünger M., and Thienel S. The ABACUS system for branch-and-cut-and-price-algorithms in integer programming and combinatorial optimization. Software: Practice & Experience 30 11 (2000) 1325-1352
33. n. SIAM Journal on Discrete Mathematics 20 1 (2006) 189-202
34. de Klerk E., Pasechnik D.V., and Schrijver A. Reduction of symmetric semidefinite programs using the regular *-representation. Mathematical Programming 109 2 (2007) 613-624
35. Kratochvíl J. String graphs. II.: Recognizing string graphs is NP-hard. Journal of Combinatorial Theory, Series B 52 1 (1991) 67-78
36. Kuratowski C. Sur le problème des courbes gauches en topologie. Fundamenta Mathematicae 15 (1930) 271-283
37. J.A. La Poutré, Alpha-algorithms for incremental planarity testing, in: Proceedings of the 26th Annual ACM Symposium on the Theory of Computing, 1994, pp. 706-715
38. Liu P., and Geldmacher R. On the deletion of nonplanar edges of a graph. Proceedings of the 10th Southeastern Conference on Combinatorics, Graph Theory, and Computing (1977), Boca Raton, FL 727-738
39. S. Masuda, T. Kashiwabara, K. Nakajima, T. Fujisawa, On the NP-completeness of a computer network layout problem, in: Proceedings of the IEEE International Symposium on Circuits and Systems, 1987, pp. 292-295
40. Masuda S., Nakajima K., Kashiwabara T., and Fujisawa T. Crossing minimization in linear embeddings of graphs. IEEE Transactions on Computers 39 1 (1990) 124-127
41. Mutzel P., and Ziegler T. The constrained crossing minimization problem. In: Kratochvil J. (Ed). Graph Drawing '99. Lecture Notes in Computer Science vol. 1731 (1999), Springer-Verlag 175-185
42. OGDF - Open Graph Drawing Framework. University of Dortmund, Chair of Algorithm Engineering and Systems Analysis. http://www.ogdf.net
43. Pach J., and Tóth G. Graphs drawn with few crossings per edge. Combinatorica 17 3 (1997) 427-439
44. Purchase H.C. Which aesthetic has the greatest effect on human understanding?. In: Di Battista G. (Ed). Graph Drawing '97. Lecture Notes in Computer Science vol. 1353 (1997), Springer-Verlag 248-261
45. Turán P. A note of welcome. Journal of Graph Theory 1 (1977) 7-9
46. Zarankiewicz K. The solution of a certain problem on graphs of P. Turán. Bulletin de l'Academie Polonaise des sciences, Cl. III 1 (1953) 167-168