Advertisement

Approximating the Maximum Rectilinear Crossing Number

  • Samuel BaldEmail author
  • Matthew P. Johnson
  • Ou Liu
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9797)

Abstract

Drawing a graph in a way that minimizes the number of edge-crossings is a well-studied problem. Recently there has been work characterizing both the minimum and maximum number of edge-crossings possible in various graph classes, assuming rectilinear (straight-line) edges. In this paper, we investigate the algorithmic problem of maximizing the number of edge-crossings over all rectilinear drawings a graph. We show that this problem is NP-hard and lies in \(\exists \mathbb {R}\). We give a nontrivial derandomization of the natural randomized 1/3-approximation algorithm, which generalizes to a weighted setting as well as to an ordering constraint satisfaction problem. We evaluate these algorithms and other heuristics in simulation.

Notes

Acknowledgements

This work was supported in part by PSC-CUNY Research Award 67665-00 45, CUNY Collaborative Incentive Research Grant (CIRG 21) 2153, and a Research in the Classroom Idea Grant.

References

  1. 1.
    Alon, N., Spencer, J.: The Probabilistic Method, chap. 15, pp. 249–258. Wiley, Hoboken (1992)Google Scholar
  2. 2.
    Alpert, M., Feder, E., Harborth, H.: The maximum of the maximum rectilinear crossing numbers of \(d\)-regular graphs of order \(n\). Electr. J. Comb. 16(1) (2009)Google Scholar
  3. 3.
    Bienstock, D.: Some provably hard crossing number problems. Discrete Comput. Geom. 6(5), 443–459 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Cabello, S., Mohar, B.: Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM J. Comput. 42(5), 1803–1829 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Canny, J.: Some algebraic and geometric computations in pspace. In: Proceedings of the Twentieth Annual ACM Symposium on Theory of Computing, STOC 1988, pp. 460–467. ACM, New York (1988)Google Scholar
  6. 6.
    Chuzhoy, J.: An algorithm for the graph crossing number problem. CoRR, abs/1012.0255 (2010)Google Scholar
  7. 7.
    Erdős, P., Guy, R.K.: Crossing number problems. Am. Math. Mon. 80, 52–58 (1973)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Erdős, P., Rényi, A.: On random graphs. i. Publicationes Math. 6, 290–297 (1959)zbMATHGoogle Scholar
  9. 9.
    Feder, E., Harborth, H., Herzberg, S., Klein, S.: The maximum rectilinear crossing number of the Petersen graph. Congr. Numerantium 206, 31–40 (2010)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Furry, W., Kleitman, D.: Maximal rectilinear crossings of cycles. Stud. Appl. Math. 56, 159–167 (1977)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Garey, M.R., Johnson, D.S.: Crossing number is NP-complete. SIAM J. Algebraic Discrete Methods 4(3), 312–316 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Guruswami, V., Hstad, J., Manokaran, R., Raghavendra, P., Charikar, M.: Beating the random ordering is hard: every ordering CSP is approximation resistant. SIAM J. Comput. 40(3), 878–914 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Harborth, H.: Drawing of the cycle graph. Congr. Numer. 66, 15–22 (1988)MathSciNetGoogle Scholar
  14. 14.
    Hlinéný, P.: Crossing number is hard for cubic graphs. J. Comb. Theory Ser. B 96(4), 455–471 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Kang, M., Pikhurko, O., Ravsky, A., Schacht, M., Verbitsky, O.: Obfuscated drawings of planar graphs. CoRR, abs/0803.0858 (2008)Google Scholar
  16. 16.
    Pach, J., Ábrego, B., Fernández-Merchant, S., Salazar, G.: The rectilinear crossing number of \(K_n\): closing in (or are we?). In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 5–18. Springer, New York (2013)CrossRefGoogle Scholar
  17. 17.
    Pach, J., Tóth, G.: Thirteen problems on crossing numbers. Geombinatorics 9, 194–207 (2000)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Pach, J., Tóth, G.: Which crossing number is it anyway? J. Comb. Theory, Ser. B 80(2), 225–246 (2000)Google Scholar
  19. 19.
    Ringel, G.: Extremal problems in the theory of graphs. In: Fiedler, M. (ed.) Theory of Graphs and Its Applications, Proceedings of Symposium Smolenice 1963, Prague, pp. 85–90 (1964)Google Scholar
  20. 20.
    Schaefer, M.: Complexity of some geometric and topological problems. In: Eppstein, D., Gansner, E.R. (eds.) GD 2009. LNCS, vol. 5849, pp. 334–344. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  21. 21.
    Schaefer, M.: The graph crossing number, its variants: a survey. Electron. J. Combin. Dyn. Surv. 21 (2014)Google Scholar
  22. 22.
    Turán, P.: A note of welcome. J. Graph Theory 1(1), 7–9 (1977)CrossRefGoogle Scholar
  23. 23.
    Verbitsky, O.: On the obfuscation complexity of planar graphs. Theor. Comput. Sci. 396(1–3), 294–300 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Weisstein, E.W.: Star polygon (2010)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.The Graduate Center of the City University of New YorkNew YorkUSA
  2. 2.Lehman CollegeCity University of New YorkNew YorkUSA

Personalised recommendations