research-article

Experiments on exact crossing minimization using column generation

Authors:

Markus Chimani,

Carsten Gutwenger,

Petra MutzelAuthors Info & Claims

Journal of Experimental Algorithmics (JEA), Volume 14

Article No.: 4, Pages 3.4 - 3.18

https://doi.org/10.1145/1498698.1564504

Published: 05 January 2010 Publication History

Abstract

The crossing number of a graph G is the smallest number of edge crossings in any drawing of G into the plane. Recently, the first branch-and-cut approach for solving the crossing number problem has been presented in Buchheim et al. [2005]. Its major drawback was the huge number of variables out of which only very few were actually used in the optimal solution. This restricted the algorithm to rather small graphs with low crossing number.

In this article, we discuss two column generation schemes; the first is based on traditional algebraic pricing, and the second uses combinatorial arguments to decide whether and which variables need to be added. The main focus of this article is the experimental comparison between the original approach and these two schemes. In addition, we evaluate the quality achieved by the best-known crossing number heuristic by comparing the new results with the results of the heuristic.

References

[1]

Batini, C., Talamo, M., and Tamassia, R. 1984. Computer aided layout of entity-relationship diagrams. J. Syst. Softw. 4, 163--173.

Digital Library

[2]

Buchheim, C., Chimani, M., Ebner, D., Gutwenger, C., J&#252;nger, M., Klau, G. W., Mutzel, P., and Weiskircher, R. 2008. A branch-and-cut approach to the crossing number problem. Discrete Optimization 5, 2, 373--388.

Digital Library

[3]

Buchheim, C., Ebner, D., J&#252;nger, M., Klau, G. W., Mutzel, P., and Weiskircher, R. 2005. Exact crossing minimization. In Proceedings of the 13th Annual Symposium on Graph Drawing 2005. Springer-Verlag, Berlin, 37--48.

Digital Library

[4]

Cai, J., Han, X., and Tarjan, R. E. 1993. An O(mlog n)-time algorithm for the maximal planar subgraph problem. SIAM J. Comput. 22, 6, 1142--1162.

Digital Library

[5]

C&#462;linescu, G., Fernandes, C. G., Finkler, U., and Karloff, H. 1998. A better approximation algorithm for finding planar subgraphs. J. Algorithms 27, 2, 269--302.

Digital Library

[6]

Chimani, M. 2008. Computing crossing numbers. Ph.D. thesis, Chair for Algorithm Engineering, Department of Computer Science, University of Dortmund, Germany.

[7]

Dantzig, G. B. and Wolfe, P. 1960. Decomposition principle for linear programs. Oper. Res. 8, 101--111.

Digital Library

[8]

Di Battista, G., Eades, P., Tamassia, R., and Tollis, I. 1999. Graph Drawing. Prentice Hall, Upper Saddle River, NJ.

[9]

Di Battista, G., Garg, A., Liotta, G., Tamassia, R., Tassinari, E., and Vargiu, F. 1997. An experimental comparison of four graph drawing algorithms. Comput. Geom. 7, 5-6, 303--325.

Digital Library

[10]

Djidjev, H. N. 1995. A linear algorithm for the maximal planar subgraph problem. In Proceedings of the 4th Workshop on Algorithms and Data Structures. Springer-Verlag, Berlin, 369--380.

Digital Library

[11]

Garey, M. R. and Johnson, D. S. 1979. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, New York.

Digital Library

[12]

Gutwenger, C. and Chimani, M. 2005. Non-planar core reduction of graphs. In Proceedings of the 13<sup>th</sup> Annual Symposium on Graph Drawing 2005. Springer-Verlag, Berlin, 223--234.

Digital Library

[13]

Gutwenger, C. and Mutzel, P. 2004. An experimental study of crossing minimization heuristics. In Proceedings of the 11th Symposium on Graph Drawing 2003. Springer-Verlag, Berlin,13--24.

[14]

Gutwenger, C., Mutzel, P., and Weiskircher, R. 2005. Inserting an edge into a planar graph. Algorithmica 41, 4, 289--308.

[15]

Hsu, W.-L. 2005. A linear time algorithm for finding a maximal planar subgraph based on PC-trees. In Proceedings of the Annual International Conference on Computing and Combinatorics. Springer, Berlin, 787--797.

[16]

Jayakumar, R., Thulasiraman, K., and Swamy, M. N. S. 1989. O(n2) algorithms for graph planarization. IEEE Trans.Comput. Aided Des. 8, 257--267.

[17]

J&#252;nger, M., Leipert, S., and Mutzel, P. 1998. A note on computing a maximal planar subgraph using PQ-trees. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 17, 7, 609--612.

Digital Library

[18]

J&#252;nger, M. and Mutzel, P. 1996. Maximum planar subgraphs and nice embeddings: Practical layout tools. Algorithmica 16, 1, 33--59.

[19]

J&#252;nger, M. and Thienel, S. 2000. The ABACUS system for branch-and-cut-and-price algorithms in integer programming and combinatorial optimization. Software: Practice & Experience 30, 11, 1325--1352.

Digital Library

[20]

Kratochv&#237;l, J. 1991. String graphs. II.: Recognizing string graphs is NP-hard. J. Comb. Theor., Series B 52, 1, 67--78.

Digital Library

[21]

Liu, P. and Geldmacher, R. 1977. On the deletion of nonplanar edges of a graph. In Proceedings of the 10th Southeastern Conference on Combinatorics, Graph Theory, and Computing. ACM, New York, 727--738.

[22]

Mutzel, P. and Ziegler, T. 1999. The constrained crossing minimization problem. In Proceedings of the Graph Drawing '99, J. Kratochvil, Ed. LNCS, vol. 1731. Springer-Verlag, Berlin, 175--185.

[23]

OGDF 2006. OGDF -- Open Graph Drawing Framework. www.ogdf.net.

[24]

Purchase, H. C. 1997. Which aesthetic has the greatest effect on human understanding&quest; In Proceedings of the 5th Annual Symposium on Graph Drawing '97 Springer-Verlag, Berlin, 248--261.

Digital Library

[25]

Tur&#225;n, P. 1977. A note of welcome. J. Graph Theor. 1, 7--9.

[26]

Vrto, I. 2006. Crossing numbers of graphs: A bibliography. ftp://ftp.ifi.savba.sk/pub/imrich/crobib.pdf.

[27]

Williamson, S. G. 1984. Depth-first search and Kuratowski subgraphs. JACM 31, 4, 681--693.

Digital Library

Cited By

Staš MFortes JŠvecová M(2024)From the Crossing Numbers of K5 + Pn and K5 + Cn to the Crossing Numbers of Wm + Sn and Wm + WnAxioms10.3390/axioms1307042713:7(427)Online publication date: 25-Jun-2024
https://doi.org/10.3390/axioms13070427
di Bartolomeo SRiedewald MGatterbauer WDunne C(2022)STRATISFIMAL LAYOUT: A modular optimization model for laying out layered node-link network visualizationsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2021.311475628:1(324-334)Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.1109/TVCG.2021.3114756
Klešč MStaš MPetrillová J(2022)The Crossing Numbers of Join of Special Disconnected Graph on Five Vertices with Discrete GraphsGraphs and Combinatorics10.1007/s00373-021-02423-538:2Online publication date: 1-Feb-2022
https://doi.org/10.1007/s00373-021-02423-5
Show More Cited By

Index Terms

Experiments on exact crossing minimization using column generation
1. Mathematics of computing
  1. Discrete mathematics
    1. Combinatorics
      1. Combinatorial algorithms
    2. Graph theory
      1. Graph algorithms
  2. Mathematical analysis
    1. Mathematical optimization
      1. Continuous optimization
        Linear programming
      2. Mixed discrete-continuous optimization
        Integer programming
2. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization
      1. Continuous optimization
        Linear programming
      2. Mixed discrete-continuous optimization
        Integer programming

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The k-planar crossing number of a graph is the minimum number of crossings of its edges over all possible drawings of the graph in k planes. We propose algorithms and methods for k-planar drawings of general graphs together with lower bound techniques. ...

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Published In

cover image ACM Journal of Experimental Algorithmics

ACM Journal of Experimental Algorithmics Volume 14, Issue

2009

613 pages

ISSN:1084-6654

EISSN:1084-6654

DOI:10.1145/1498698

Issue’s Table of Contents

Copyright © 2010 ACM.

Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

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Association for Computing Machinery

New York, NY, United States

Publication History

Published: 05 January 2010

Accepted: 01 March 2009

Revised: 01 February 2009

Received: 01 September 2006

Published in JEA Volume 14

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Cited By

Staš MFortes JŠvecová M(2024)From the Crossing Numbers of K5 + Pn and K5 + Cn to the Crossing Numbers of Wm + Sn and Wm + WnAxioms10.3390/axioms1307042713:7(427)Online publication date: 25-Jun-2024
https://doi.org/10.3390/axioms13070427
di Bartolomeo SRiedewald MGatterbauer WDunne C(2022)STRATISFIMAL LAYOUT: A modular optimization model for laying out layered node-link network visualizationsIEEE Transactions on Visualization and Computer Graphics10.1109/TVCG.2021.311475628:1(324-334)Online publication date: 1-Jan-2022
https://dl.acm.org/doi/10.1109/TVCG.2021.3114756
Klešč MStaš MPetrillová J(2022)The Crossing Numbers of Join of Special Disconnected Graph on Five Vertices with Discrete GraphsGraphs and Combinatorics10.1007/s00373-021-02423-538:2Online publication date: 1-Feb-2022
https://doi.org/10.1007/s00373-021-02423-5
Chimani MIlsen MWiedera T(2021)Star-Struck by Fixed Embeddings: Modern Crossing Number HeuristicsGraph Drawing and Network Visualization10.1007/978-3-030-92931-2_3(41-56)Online publication date: 14-Sep-2021
https://dl.acm.org/doi/10.1007/978-3-030-92931-2_3

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