Approximation Algorithms for Intersection Graphs

  • Frank Kammer
  • Torsten Tholey
  • Heiko Voepel
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6302)


We study three complexity parameters that in some sense measure how chordal-like a graph is. The similarity to chordal graphs is used to construct simple polynomial-time approximation algorithms with constant approximation ratio for many \(\mathcal{NP}\)-hard problems, when restricted to graphs for which at least one of the three complexity parameters is bounded by a constant. As applications we present approximation algorithms with constant approximation ratio for maximum weighted independent set, minimum (independent) dominating set, minimum vertex coloring, maximum weighted clique, and minimum clique partition for large classes of intersection graphs.


Approximation Algorithm Complexity Parameter Maximum Clique Intersection Graph Chordal Graph 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Akcoglu, K., Aspnes, J., DasGupta, B., Kao, M.-Y.: Opportunity cost algorithms for combinatorial auctions. In: Kontoghiorghes, E.J., Rustem, B., Siokos, S. (eds.) Applied Optimization: Computational Methods in Decision Making, Economics and Finance, vol. 74, pp. 455–479 (2002)Google Scholar
  2. 2.
    Bar-Yehuda, R., Halldórsson, M.M., Naor, J., Shachnai, H., Shapira, I.: Scheduling split intervals. SIAM J. Comput. 36, 1–15 (2006)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996)MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Butman, A., Hermelin, D., Lewenstein, M., Rawitz, D.: Optimization problems in multiple-interval graphs. In: Proc.18th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2007), pp. 268–277 (2007)Google Scholar
  5. 5.
    Cerioli, M.R., Faria, L., Ferreira, T.O., Protti, F.: On minimum clique partition and maximum independent set on unit disk graphs and penny graphs: complexity and approximation. Electronic Notes in Discrete Mathematics 18, 73–79 (2004)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Chan, T.M.: Polynomial-time approximation schemes for packing and piercing fat objects. J. Algorithms 46, 178–189 (2003)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math. 86, 165–177 (1990)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inform. and Comput. 85, 12–75 (1990)MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Dumitrescu, A., Pach, J.: Minimum clique partition in unit disk graphs, arXiv:0909.1552v1Google Scholar
  10. 10.
    Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time approximation schemes for geometric intersection graphs. In: Proc. 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), pp. 671–679 (2001)Google Scholar
  11. 11.
    Erlebach, T., van Leeuwen, E.J.: Domination in geometric intersection graphs. In: Laber, E.S., Bornstein, C., Nogueira, L.T., Faria, L. (eds.) LATIN 2008. LNCS, vol. 4957, pp. 747–758. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  12. 12.
    Fowler, R.J., Paterson, M.S., Tanimoto, S.L.: Optimal packing and covering in the plane are NP-complete. Inform. Process. Lett. 12, 133–137 (1981)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Frank, A.: Some polynomial algorithms for certain graphs and hypergraphs. In: Proc. 5th British Combinatorial Conference (Aberdeen 1975), Congr. Numer. XV., pp. 211–226 (1976)Google Scholar
  14. 14.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete graph problems. Theoret. Comput. Sci. 1, 237–267 (1976)MATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Gavril, F.: Algorithms for minimum coloring, maximum clique, minimum covering by cliques, and maximum independent set of a chordal graph. SIAM J. Comput. 1, 180–187 (1972)MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Gibson, T., Pirwani, I.A.: Approximation algorithms for dominating set in disk graphs. In: Proc. 18th Annual European Symposium on Algorithms (ESA 2010). LNCS. Springer, Heidelberg (2010) (to appear)Google Scholar
  17. 17.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)MATHGoogle Scholar
  18. 18.
    Golumbic, M.C.: Algorithmic aspects of intersection graphs and representation hypergraphs. Graphs and Combinatorics 4, 307–321 (1988)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Gräf, A.: Coloring and recognizing special graph classes, Technical Report Musikinformatik und Medientechnik Bericht 20/95, Johannes Gutenberg-Universität Mainz (1995)Google Scholar
  20. 20.
    Gramm, J., Guo, J., Hüffner, F., Niedermeier, R.: Data reduction and exact algorithms for clique cover. Journal of Experimental Algorithmics 13, Article No. 2 (2009)Google Scholar
  21. 21.
    Griggs, J.R., West, D.B.: Extremal values of the interval number of a graph. SIAM Journal on algebraic and discrete methods 1, 1–7 (1980)MATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Hunt III, H.B., Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: NC-pproximation schemes for NP- and PSPACE-hard problems for geometric graphs. J. Algorithms 26, 238–274 (1998)MATHCrossRefMathSciNetGoogle Scholar
  23. 23.
    Hurink, J.L., Nieberg, T.: Approximating minimum independent dominating sets in wireless networks. Inform. Process. Lett. 109, 155–160 (2008)CrossRefMathSciNetGoogle Scholar
  24. 24.
    Imai, H., Asano, T.: Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane. J. Algorithms 4, 310–323 (1983)MATHCrossRefMathSciNetGoogle Scholar
  25. 25.
    Jamison, R.E., Mulder, H.M.: Tolerance intersection graphs on binary trees trees with constant tolerance 3. Discrete Math. 215, 115–131 (2000)MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Johnson, D.S.: Approximation algorithms for combinatorial problems. J. Comput. System Sci. 9, 256–278 (1974)MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Kammer, F., Tholey, T., Voepel, H.: Approximation algorithms for intersection graphs, Report 2009–6, Institut für Informatik, Universität Augsburg (2009)Google Scholar
  28. 28.
    Malesińska, E.: Graph-theoretical models for frequency assignment problems, PhD thesis, University of Berlin (1997)Google Scholar
  29. 29.
    Marathe, M.V., Breu, H., Hunt III, H.B., Ravi, S.S., Rosenkrantz, D.J.: Simple heuristics for unit disk graphs. Networks 25, 59–68 (1995)MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Nieberg, T., Hurink, J., Kern, W.: Approximation Schemes for Wireless Networks. ACM Transactions on Algorithms 4, Article No. 49 (2008)Google Scholar
  31. 31.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. System Sci. 43, 425–440 (1991)MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    Pirwani, I.A., Salavatipour, M.R.: A weakly-robust PTAS for minimum clique partition on unit disk graphs. In: Kaplan, H. (ed.) SWAT 2010. LNCS, vol. 6139, pp. 188–199. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  33. 33.
    Tardos, É.: A strongly polynomial algorithm to solve combinatorial linear programs. Operations Research 34, 250–256 (1986)MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Ye, Y., Borodin, A.: Elimination graphs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5556, pp. 774–785. Springer, Heidelberg (2009)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Frank Kammer
    • 1
  • Torsten Tholey
    • 1
  • Heiko Voepel
    • 1
  1. 1.Institut für InformatikUniversität AugsburgAugsburgGermany

Personalised recommendations