Improved Approximation Algorithms for Box Contact Representations

  • Michael A. Bekos
  • Thomas C. van Dijk
  • Martin Fink
  • Philipp Kindermann
  • Stephen Kobourov
  • Sergey Pupyrev
  • Joachim Spoerhase
  • Alexander Wolff
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8737)

Abstract

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 show that the problem is APX-complete on bipartite graphs of bounded maximum degree. 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 consider several planar graph classes (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.

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References

  1. 1.
    Barth, L., Fabrikant, S.I., Kobourov, S.G., 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.) LATIN 2014. LNCS, vol. 8392, pp. 514–525. Springer, Heidelberg (2014)Google Scholar
  2. 2.
    Barth, L., Kobourov, S.G., Pupyrev, S.: Experimental comparison of semantic word clouds. In: Gudmundsson, J., Katajainen, J. (eds.) SEA 2014. LNCS, vol. 8504, pp. 247–258. Springer, Heidelberg (2014)CrossRefGoogle Scholar
  3. 3.
    Bekos, M., van Dijk, T., Fink, M., Kindermann, P., Kobourov, S.G., Pupyrev, S., Spoerhase, J., Wolff, A.: Improved approximation algorithms for box contact representations. Arxiv report (2014) arxiv.org/abs/1403.4861Google Scholar
  4. 4.
    Briest, P., Krysta, P., Vöcking, B.: Approximation techniques for utilitarian mechanism design. SIAM J. Comput. 40(6), 1587–1622 (2011)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Buchsbaum, A.L., Gansner, E.R., Procopiuc, C.M., Venkatasubramanian, S.: Rectangular layouts and contact graphs. ACM Trans. Algorithms 4(1) (2008)Google Scholar
  6. 6.
    Chekuri, C., Khanna, S.: A PTAS for the multiple knapsack problem. In: 11th ACM-SIAM Symp. Discrete Algorithms (SODA), pp. 213–222. SIAM (2000)Google Scholar
  7. 7.
    Cohen, R., Katzir, L., Raz, D.: An efficient approximation for the generalized assignment problem. Inf. Process. Lett. 100(4), 162–166 (2006)CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    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
  9. 9.
    Dwyer, T., Marriott, K., Stuckey, P.J.: Fast node overlap removal. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 153–164. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  10. 10.
    Eppstein, D., Mumford, E., Speckmann, B., Verbeek, K.: Area-universal and constrained rectangular layouts. SIAM J. Comput. 41(3), 537–564 (2012)CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    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
  12. 12.
    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
  13. 13.
    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)CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Frederickson, G.N.: Fast algorithms for shortest paths in planar graphs, with applications. SIAM J. Comput. 16(6), 1004–1022 (1987)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Gansner, E.R., Hu, Y.: Efficient, proximity-preserving node overlap removal. J. Graph Algortihms Appl. 14(1), 53–74 (2010)CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Hakimi, S.L., Mitchem, J., Schmeichel, E.F.: Star arboricity of graphs. Discrete Math. 149(1-3), 93–98 (1996)CrossRefMATHMathSciNetGoogle Scholar
  17. 17.
    Li, H.: Word clustering and disambiguation based on co-occurrence data. J. Nat. Lang. Eng. 8(1), 25–42 (2002)Google Scholar
  18. 18.
    Nash-Williams, C.: Decomposition of finite graphs into forests. J. L. Math. Soc. 39, 12 (1964)MATHMathSciNetGoogle Scholar
  19. 19.
    Nishizeki, T., Baybars, I.: Lower bounds on the cardinality of the maximum matchings of planar graphs. Discrete Math. 28(3), 255–267 (1979)CrossRefMATHMathSciNetGoogle Scholar
  20. 20.
    Nöllenburg, M., Prutkin, R., Rutter, I.: Edge-weighted contact representations of planar graphs. J. Graph Algorithms Appl. 17(4), 441–473 (2013)CrossRefMATHMathSciNetGoogle Scholar
  21. 21.
    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
  22. 22.
    Raisz, E.: The rectangular statistical cartogram. Geogr. Review 24(3), 292–296 (1934)CrossRefGoogle Scholar
  23. 23.
    Viégas, F.B., Wattenberg, M., Feinberg, J.: Participatory visualization with Wordle. IEEE Trans. Vis. Comput. Graph. 15(6), 1137–1144 (2009)CrossRefGoogle Scholar
  24. 24.
    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-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Michael A. Bekos
    • 1
  • Thomas C. van Dijk
    • 2
  • Martin Fink
    • 2
    • 5
  • Philipp Kindermann
    • 2
  • Stephen Kobourov
    • 3
  • Sergey Pupyrev
    • 3
    • 4
  • Joachim Spoerhase
    • 2
  • Alexander Wolff
    • 2
  1. 1.Wilhelm-Schickard-Institut für InformatikUniversität TübingenGermany
  2. 2.Lehrstuhl für Informatik IUniversität WürzburgGermany
  3. 3.Department of Computer ScienceUniversity of ArizonaUSA
  4. 4.Institute of Mathematics and Computer ScienceUral Federal UniversityRussia
  5. 5.Department of Computer SicenceUniversity of CaliforniaSanta BarbaraUSA

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