Theory of Computing Systems

, Volume 47, Issue 3, pp 613–636 | Cite as

Trimming of Graphs, with Application to Point Labeling

  • Thomas Erlebach
  • Torben HagerupEmail author
  • Klaus Jansen
  • Moritz Minzlaff
  • Alexander Wolff
Open Access


For t>0 and g≥0, a vertex-weighted graph of total weight W is (t,g)-trimmable if it contains a vertex-induced subgraph of total weight at least (1−1/t)W and with no simple path of more than g edges. A family of graphs is trimmable if for every constant t>0, there is a constant g≥0 such that every vertex-weighted graph in the family is (t,g)-trimmable. We show that every family of graphs of bounded domino treewidth is trimmable. This implies that every family of graphs of bounded degree is trimmable if the graphs in the family have bounded treewidth or are planar. We also show that every family of directed graphs of bounded layer bandwidth (a less restrictive condition than bounded directed bandwidth) is trimmable. As an application of these results, we derive polynomial-time approximation schemes for various forms of the problem of labeling a subset of given weighted point features with nonoverlapping sliding axes-parallel rectangular labels so as to maximize the total weight of the labeled features, provided that the ratios of label heights or the ratios of label lengths are bounded by a constant. This settles one of the last major open questions in the theory of map labeling.


Trimming weighted graphs Domino treewidth Planar graphs Layer bandwidth Point-feature label placement Map labeling Sliding labels Polynomial-time approximation schemes 


  1. 1.
    Agarwal, P.K., van Kreveld, M., Suri, S.: Label placement by maximum independent set in rectangles. Comput. Geom. Theory Appl. 11, 209–218 (1998) zbMATHGoogle Scholar
  2. 2.
    Baker, B.S.: Approximation algorithms for NP-complete problems on planar graphs. J. ACM 41, 153–180 (1994) zbMATHCrossRefGoogle Scholar
  3. 3.
    Bodlaender, H.L.: A partial k-arboretum of graphs with bounded treewidth. Theor. Comput. Sci. 209(1–2), 1–45 (1998) zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Bodlaender, H.L.: A note on domino treewidth. Discrete Math. Theor. Comput. Sci. 3(4), 141–150 (1999) zbMATHMathSciNetGoogle Scholar
  5. 5.
    Ding, G., Oporowski, B.: Some results on tree decomposition of graphs. J. Graph Theory 20, 481–499 (1995) zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Duncan, R., Qian, J., Vigneron, A., Zhu, B.: Polynomial time algorithms for three-label point labeling. Theor. Comput. Sci. 296(1), 75–87 (2003) zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Garey, M.R., Graham, R.L., Johnson, D.S., Knuth, D.E.: Complexity results for bandwidth minimization. SIAM J. Appl. Math. 34(3), 477–495 (1978) zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Hochbaum, D.S., Maass, W.: Approximation schemes for covering and packing problems in image processing and VLSI. J. ACM 32, 130–136 (1985) zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Haussler, D., Welzl, E.: ε-nets and simplex range queries. Discrete Comput. Geom. 2, 127–151 (1987) zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Jiang, M.: A new approximation algorithm for labeling points with circle pairs. Inf. Process. Lett. 99(4), 125–129 (2006) zbMATHCrossRefGoogle Scholar
  11. 11.
    Poon, S.-H., Shin, C.-S., Strijk, T., Uno, T., Wolff, A.: Labeling points with weights. Algorithmica 38(2), 341–362 (2003) CrossRefMathSciNetGoogle Scholar
  12. 12.
    Qin, Z., Wolff, A., Xu, Y., Zhu, B.: New algorithms for two-label point labeling. In: Paterson, M. (ed.) Proc. 8th Annual European Symposium on Algorithms (ESA’00). Lecture Notes Comput. Sci., vol. 1879, pp. 368–379. Springer, Berlin (2000) Google Scholar
  13. 13.
    Robertson, N., Seymour, P.D.: Graph minors. II. Algorithmic aspects of tree-width. J. Algorithms 7(3), 309–322 (1986) zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    van Kreveld, M., Strijk, T., Wolff, A.: Point labeling with sliding labels. Comput. Geom. Theory Appl. 13, 21–47 (1999) zbMATHGoogle Scholar
  15. 15.
    Wolff, A., Strijk, T.: The map-labeling bibliography. (1996–2008).

Copyright information

© The Author(s) 2009

Authors and Affiliations

  • Thomas Erlebach
    • 1
  • Torben Hagerup
    • 2
    Email author
  • Klaus Jansen
    • 3
  • Moritz Minzlaff
    • 4
  • Alexander Wolff
    • 5
  1. 1.Department of Computer ScienceUniversity of LeicesterLeicesterUK
  2. 2.Institut für InformatikUniversität AugsburgAugsburgGermany
  3. 3.Institut für Informatik und Praktische MathematikUniversität KielKielGermany
  4. 4.Institut für MathematikTechnische Universität BerlinBerlinGermany
  5. 5.Faculteit Wiskunde en InformaticaTechnische Universiteit EindhovenEindhovenThe Netherlands

Personalised recommendations