Metro-Line Crossing Minimization: Hardness, Approximations, and Tractable Cases

Fink, Martin and Pupyrev, Sergey (2013) Metro-Line Crossing Minimization: Hardness, Approximations, and Tractable Cases. In: 21st International Symposium, GD 2013, September 23-25, 2013, Bordeaux, France , pp. 328-339 (Official URL:

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Crossing minimization is one of the central problems in graph drawing. Recently, there has been an increased interest in the problem of minimizing crossings between paths in drawings of graphs. This is the metro-line crossing minimization problem (MLCM): Given an embedded graph and a set L of simple paths, called lines, order the lines on each edge so that the total number of crossings is minimized. So far, the complexity of MLCM has been an open problem. In contrast, the problem variant in which line ends must be placed in outermost position on their edges (MLCM-P) is known to be NP-hard. Our main results answer two open questions: (i) We show that MLCM is NP-hard. (ii) We give an O(log|L|−−−−−−√) -approximation algorithm for MLCM-P.

Item Type:Conference Paper
Classifications:G Algorithms and Complexity > G.420 Crossings
P Styles > P.999 Others
ID Code:1386

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Agarwal, A., Charikar, M., Makarychev, K., Makarychev, Y.: O (\sqrt{log|L|}\) approximation algorithms for min UnCut, min 2CNF deletion, and directed cut problems. In: STOC 2005, pp. 573–581. ACM, New York (2005)

Argyriou, E.N., Bekos, M.A., Kaufmann, M., Symvonis, A.: On metro-line crossing minimization. Journal of Graph Algorithms and Applications 14(1), 75–96 (2010)

Asquith, M., Gudmundsson, J., Merrick, D.: An ILP for the metro-line crossing problem. In: Harland, J., Manyem, P. (eds.) CATS 2008. CRPIT, vol. 77, pp. 49–56. Australian Computer Society (2008)

Bekos, M.A., Kaufmann, M., Potika, K., Symvonis, A.: Line crossing minimization on metro maps. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 231–242. Springer, Heidelberg (2008)

Benkert, M., Nöllenburg, M., Uno, T., Wolff, A.: Minimizing intra-edge crossings in wiring diagrams and public transportation maps. In: Kaufmann, M., Wagner, D. (eds.) GD 2006. LNCS, vol. 4372, pp. 270–281. Springer, Heidelberg (2007)

Fink, M., Pupyrev, S.: Metro-line crossing minimization: Hardness, approximations, and tractable cases. ArXiv e-print abs/1306.2079 (2013),

Fink, M., Pupyrev, S.: Ordering metro lines by block crossings. In: Chatterjee, K., Sgall, J. (eds.) MFCS 2013. LNCS, vol. 8087, pp. 397–408. Springer, Heidelberg (2013)

Groeneveld, P.: Wire ordering for detailed routing. IEEE Des. Test 6(6), 6–17 (1989)

Grötschel, M., Pulleyblank, W.: Weakly bipartite graphs and the Max-Cut problem. Operations Research Letters 1(1), 23–27 (1981)

Marek-Sadowska, M., Sarrafzadeh, M.: The crossing distribution problem. IEEE Transactions on CAD of Integrated Circuits and Systems 14(4), 423–433 (1995)

Nöllenburg, M.: Network Visualization: Algorithms, Applications, and Complexity. Ph.D. thesis, Fakultät für Informatik, Universität Karlsruhe (TH) (2009)

Nöllenburg, M.: An improved algorithm for the metro-line crossing minimization problem. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, pp. 381–392. Springer, Heidelberg (2010)

Okamoto, Y., Tatsu, Y., Uno, Y.: Exact and fixed-parameter algorithms for metro-line crossing minimization problems. ArXiv e-print abs/1306.3538 (2013)

Pupyrev, S., Nachmanson, L., Bereg, S., Holroyd, A.E.: Edge routing with ordered bundles. In: van Kreveld, M.J., Speckmann, B. (eds.) GD 2011. LNCS, vol. 7034, pp. 136–147. Springer, Heidelberg (2012)

Razgon, I., O’Sullivan, B.: Almost 2-SAT is fixed-parameter tractable. Journal of Computer and System Sciences 75(8), 435–450 (2009)