Abstract
Let G be a planar graph and F a set of additional edges not yet in G. The multiple edge insertion problem (MEI) asks for a drawing of G + F with the minimum number of pairwise edge crossings, such that the subdrawing of G is plane. As an exact solution to MEI is NP-hard for general F, we present the first approximation algorithm for MEI which achieves an additive approximation factor (depending only on the size of F and the maximum degree of G) in the case of connected G. Our algorithm seems to be the first directly implementable one in that realm, too, next to the single edge insertion.
It is also known that an (even approximate) solution to the MEI problem would approximate the crossing number of the F-almost-planar graph G + F, while computing the crossing number of G + F exactly is NP-hard already when |F| = 1. Hence our algorithm induces new, improved approximation bounds for the crossing number problem of F-almost-planar graphs, achieving constant-factor approximation for the large class of such graphs of bounded degrees and bounded size of F.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Arora, S., Rao, S., Vazirani, U.: Expander flows, geometric embeddings and graph partitioning. J. ACM 56, 5:1–5:37 (2009)
Bienstock, D., Monma, C.L.: On the complexity of embedding planar graphs to minimize certain distance measures. Algorithmica 5(1), 93–109 (1990)
Cabello, S., Mohar, B.: Crossing and weighted crossing number of near-planar graphs. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 38–49. Springer, Heidelberg (2009)
Cabello, S., Mohar, B.: Adding one edge to planar graphs makes crossing number hard. In: Proc. SoCG 2010, pp. 68–76. ACM, New York (2010)
Chimani, M.: Computing Crossing Numbers. PhD thesis, TU Dortmund, Germany (2008), http://www.ae.uni-jena.de/alenmedia/dokumente/ComputingCrossingNumbers_PhDthesis_Chimani_pdf.pdf
Chimani, M., Gutwenger, C., Mutzel, P., Wolf, C.: Inserting a vertex into a planar graph. In: Proc. SODA 2009, pp. 375–383 (2009)
Chimani, M., Hliněný, P.: A tighter insertion-based approximation of the crossing number. Full version. arXiv:1104.5039 (2011)
Chimani, M., Hliněný, P., Mutzel, P.: Approximating the crossing number of apex graphs. In: Tollis, I.G., Patrignani, M. (eds.) GD 2008. LNCS, vol. 5417, pp. 432–434. Springer, Heidelberg (2009)
Chuzhoy, J.: An algorithm for the graph crossing number problem. In: Proc. STOC 2011 (to appear, 2011)
Chuzhoy, J., Makarychev, Y., Sidiropoulos, A.: On graph crossing number and edge planarization. In: Proc. SODA 2011, pp. 1050–1069. ACM Press, New York (2011)
Di Battista, G., Tamassia, R.: On-line planarity testing. SIAM Journal on Computing 25, 956–997 (1996)
Even, G., Guha, S., Schieber, B.: Improved approximations of crossings in graph drawings and vlsi layout areas. SIAM J. Comput. 32(1), 231–252 (2002)
Gitler, I., Hliněný, P., Leanos, J., Salazar, G.: The crossing number of a projective graph is quadratic in the face-width. Electronic Notes in Discrete Mathematics 29, 219–223 (2007)
Gutwenger, C.: Application of SPQR-Trees in the Planarization Approach for Drawing Graphs. PhD thesis, TU Dortmund, Germany (2010)
Gutwenger, C., Mutzel, P.: A linear time implementation of SPQR-trees. In: Marks, J. (ed.) GD 2000. LNCS, vol. 1984, pp. 77–90. Springer, Heidelberg (2001)
Gutwenger, C., Mutzel, P.: An experimental study of crossing minimization heuristics. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 13–24. Springer, Heidelberg (2004)
Gutwenger, C., Mutzel, P., Weiskircher, R.: Inserting an edge into a planar graph. Algorithmica 41(4), 289–308 (2005)
Hliněný, P., Chimani, M.: Approximating the crossing number of graphs embeddable in any orientable surface. In: Proc. SODA 2010, pp. 918–927 (2010)
Hliněný, P., Salazar, G.: On the crossing number of almost planar graphs. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 162–173. Springer, Heidelberg (2007)
Hliněný, P., Salazar, G.: Approximating the crossing number of toroidal graphs. In: Tokuyama, T. (ed.) ISAAC 2007. LNCS, vol. 4835, pp. 148–159. Springer, Heidelberg (2007)
Hopcroft, J.E., Tarjan, R.E.: Dividing a graph into triconnected components. SIAM Journal on Computing 2(3), 135–158 (1973)
Tutte, W.T.: Connectivity in graphs. Mathematical Expositions, vol. 15. University of Toronto Press (1966)
Vrt’o, I.: Crossing numbers of graphs: A bibliography (2011), ftp://ftp.ifi.savba.sk/pub/imrich/crobib.pdf
Ziegler, T.: Crossing Minimization in Automatic Graph Drawing. PhD thesis, Saarland University, Germany (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chimani, M., Hliněný, P. (2011). A Tighter Insertion-Based Approximation of the Crossing Number. In: Aceto, L., Henzinger, M., Sgall, J. (eds) Automata, Languages and Programming. ICALP 2011. Lecture Notes in Computer Science, vol 6755. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22006-7_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-22006-7_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-22005-0
Online ISBN: 978-3-642-22006-7
eBook Packages: Computer ScienceComputer Science (R0)