, Volume 77, Issue 3, pp 902–920 | Cite as

Improved Approximation Algorithms for Box Contact Representations

  • Michael A. Bekos
  • Thomas C. van Dijk
  • Martin Fink
  • Philipp KindermannEmail author
  • Stephen Kobourov
  • Sergey Pupyrev
  • Joachim Spoerhase
  • Alexander Wolff


We study the following geometric representation problem: Given a graph whose vertices correspond to axis-aligned rectangles with fixed dimensions, arrange the rectangles without overlaps in the plane such that two rectangles touch if the graph contains an edge between them. This problem is called Contact Representation of Word Networks (Crown) since it formalizes the geometric problem behind drawing word clouds in which semantically related words are close to each other. Crown is known to be NP-hard, and there are approximation algorithms for certain graph classes for the optimization version, Max-Crown, in which realizing each desired adjacency yields a certain profit. We present the first O(1)-approximation algorithm for the general case, when the input is a complete weighted graph, and for the bipartite case. Since the subgraph of realized adjacencies is necessarily planar, we also consider several planar graph classes (namely stars, trees, outerplanar, and planar graphs), improving upon the known results. For some graph classes, we also describe improvements in the unweighted case, where each adjacency yields the same profit. Finally, we show that the problem is APX-complete on bipartite graphs of bounded maximum degree.


Word clouds Box contact representations Approximation algorithms 



We thank the anonymous reviewers for helping us to improve the presentation of our paper. We particularly thank the reviewer who contributed the idea to derandomize our algorithms for the general weighted case using Theorem 4.


  1. 1.
    Ackerman, E.: A note on 1-planar graphs. Discrete Appl. Math. 175, 104–108 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Alon, N., Spencer, J.: The Probabilistic Method. Wiley, Hoboken (1992)zbMATHGoogle Scholar
  3. 3.
    Barth, L., Fabrikant, S.I., Kobourov, S., Lubiw, A., Nöllenburg, M., Okamoto, Y., Pupyrev, S., Squarcella, C., Ueckerdt, T., Wolff, A.: Semantic word cloud representations: hardness and approximation algorithms. In: Pardo, A., Viola, A. (eds) Proceedings of 11th Latin American Symposium Theoritical Information (LATIN’14), vol 8392 of Lecture Notes in Computer Science, pp 514–525. Springer, Heidelberg (2014)Google Scholar
  4. 4.
    Barth, L., Kobourov, S., Pupyrev, S.: Experimental comparison of semantic word clouds. In: Gudmundsson, J., Katajainen, J. (eds) Proceedings of 13th International Symposium on Experimental Algorithms (SEA’14), vol 8504 of Lecture Notes in Computer Science, pp 247–258. Springer, Heidelberg (2014)Google Scholar
  5. 5.
    Briest, P., Krysta, P., Vöcking, B.: Approximation techniques for utilitarian mechanism design. SIAM J. Comput. 40(6), 1587–1622 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Buchsbaum, A.L., Gansner, E.R., Procopiuc, C.M., Venkatasubramanian, S.: Rectangular layouts and contact graphs. ACM Trans. Algorithms 4(1) (2008)Google Scholar
  7. 7.
    Chekuri, C., Khanna, S.: A PTAS for the multiple knapsack problem. In: Proceedings of 11th Annual ACM-SIAM Symposum Discrete Algorithms (SODA’00), pp 213–222. SIAM (2000)Google Scholar
  8. 8.
    Cohen, R., Katzir, L., Raz, D.: An efficient approximation for the generalized assignment problem. Inf. Process. Lett. 100(4), 162–166 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Cui, W., Wu, Y., Liu, S., Wei, F., Zhou, M., Qu, H.: Context-preserving dynamic word cloud visualization. IEEE Comput. Graph. Appl. 30(6), 42–53 (2010)CrossRefGoogle Scholar
  10. 10.
    Dwyer, T., Marriott, K., Stuckey, P.J.: Fast node overlap removal. In: Healy, P., Nikolov, N.S. (eds), Proc. 13th International Symposium on Graph Drawing (GD’05), vol 3843 of Lecture Notes in Computer Science, pp. 153–164. Springer, Heidelberg (2005)Google Scholar
  11. 11.
    Eppstein, D., Mumford, E., Speckmann, B., Verbeek, K.: Area-universal and constrained rectangular layouts. SIAM J. Comput. 41(3), 537–564 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Erkan, G., Radev, D.R.: Lexrank: graph-based lexical centrality as salience in text summarization. J. Artif. Int. Res. 22(1), 457–479 (2004)Google Scholar
  13. 13.
    Felsner, S.: Rectangle and square representations of planar graphs. In: Pach, J. (ed.) Thirty Essays on Geometric Graph Theory, pp. 213–248. Springer, Heidelberg (2013)CrossRefGoogle Scholar
  14. 14.
    Fleischer, L., Goemans, M.X., Mirrokni, V., Sviridenko, M.: Tight approximation algorithms for maximum separable assignment problems. Math. Oper. Res. 36(3), 416–431 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Frederickson, G.N.: Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Comput. 16(6), 1004–1022 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gansner, E.R., Hu, Y.: Efficient, proximity-preserving node overlap removal. J. Graph Algorithms Appl. 14(1), 53–74 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Hakimi, S.L., Mitchem, J., Schmeichel, E.F.: Star arboricity of graphs. Discrete Math. 149(1–3), 93–98 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Li, H.: Word clustering and disambiguation based on co-occurrence data. J. Nat. Lang. Eng. 8(1), 25–42 (2002)Google Scholar
  19. 19.
    Nash-Williams, C.: Decomposition of finite graphs into forests. J. Lond. Math. Soc. 39, 12 (1964)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Nishizeki, T., Baybars, I.: Lower bounds on the cardinality of the maximum matchings of planar graphs. Discrete Math. 28(3), 255–267 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Nöllenburg, M., Prutkin, R., Rutter, I.: Edge-weighted contact representations of planar graphs. J. Graph Algorithms Appl. 17(4), 441–473 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Paulovich, F.V., Toledo, F.M.B., Telles, G.P., Minghim, R., Nonato, L.G.: Semantic wordification of document collections. Comput. Graph. Forum 31(3), 1145–1153 (2012)CrossRefGoogle Scholar
  23. 23.
    Raisz, E.: The rectangular statistical cartogram. Geogr. Rev. 24(3), 292–296 (1934)CrossRefGoogle Scholar
  24. 24.
    Viégas, F.B., Wattenberg, M., Feinberg, J.: Participatory visualization with Wordle. IEEE Trans. Visual. Comput. Graphics 15(6), 1137–1144 (2009)CrossRefGoogle Scholar
  25. 25.
    Weiland, S.: Der Koalitionsvertrag im Schnellcheck (Quick overview of the [German] coalition agreement). Spiegel Online, Click on “Fotos”, 27 Nov. 2013
  26. 26.
    Wu, Y., Provan, T., Wei, F., Liu, S., Ma, K.-L.: Semantic-preserving word clouds by seam carving. Comput. Graph. Forum 30(3), 741–750 (2011)CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenTübingenGermany
  2. 2.Lehrstuhl für Informatik IUniversität WürzburgWürzburgGermany
  3. 3.Department of Computer ScienceUniversity of CaliforniaSanta BarbaraUSA
  4. 4.Department of Computer ScienceUniversity of ArizonaTucsonUSA
  5. 5.Institute of Mathematics and Computer ScienceUral Federal UniversityYekaterinburgRussia

Personalised recommendations