Algorithmica

, Volume 68, Issue 2, pp 312–336 | Cite as

Approximation Algorithms for Intersection Graphs

Article

Abstract

We study three complexity parameters that, for each vertex v, are an upper bound for the number of cliques that are sufficient to cover a subset S(v) of its neighbors. We call a graph k-perfectly groupable if S(v) consists of all neighbors, k-simplicial if S(v) consists of the neighbors with a higher number after assigning distinct numbers to all vertices, and k-perfectly orientable if S(v) consists of the endpoints of all outgoing edges from v for an orientation of all edges. These parameters measure in some sense how chordal-like a graph is—the last parameter was not previously considered in literature. The similarity to chordal graphs is used to construct simple polynomial-time approximation algorithms with constant approximation ratio for many 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.

Keywords

Intersection graphs Parameterized approximation algorithms Graph algorithms NP-hard problems 

Notes

Acknowledgements

The authors thank the anonymous reviewers for their valuable comments and suggestions that helped to improve the readability of the paper. In particular, we are grateful for the hint that the approximation ratio of the minimum clique partition problem can be improved from O(klog2 n) shown in the conference version of this paper to O(klogn) as shown in this paper.

References

  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.) Computational Methods in Decision-Making, Economics and Finance. Applied Optimization, vol. 74, pp. 455–479. Kluwer Academic, Dordrecht (2002) CrossRefGoogle 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) CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Bodlaender, H.L.: A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput. 25, 1305–1317 (1996) CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Butman, A., Hermelin, D., Lewenstein, M., Rawitz, D.: Optimization problems in multiple-interval graphs. ACM Trans. Algorithms 6, Article 40 (2010) CrossRefMathSciNetGoogle 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. Electron. Notes Discrete Math. 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) CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Clark, B.N., Colbourn, C.J., Johnson, D.S.: Unit disk graphs. Discrete Math. 86, 165–177 (1990) CrossRefMATHMathSciNetGoogle Scholar
  8. 8.
    Courcelle, B.: The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Inf. Comput. 85, 12–75 (1990) CrossRefMATHMathSciNetGoogle Scholar
  9. 9.
    Downey, R.G., Fellows, M.R.: Parameterized Complexity. Springer, New York (1999) CrossRefGoogle Scholar
  10. 10.
    Dumitrescu, A., Pach, J.: Minimum clique partition in unit disk graphs. Graphs Comb. 27, 399–411 (2011) CrossRefMATHMathSciNetGoogle Scholar
  11. 11.
    Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time approximation schemes for geometric intersection graphs. SIAM J. Comput. 34, 1302–1323 (2005) CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Erlebach, T., van Leeuwen, E.J.: Domination in geometric intersection graphs. In: Proc. 8th Latin American Theoretical Informatics Symposium (LATIN 2008). LNCS, vol. 4957, pp. 747–758. Springer, Berlin (2008) CrossRefGoogle Scholar
  13. 13.
    Feige, U.: Randomized graph products, chromatic numbers, and the Lovász ϑ function. Combinatorica 17, 79–90 (1997) CrossRefMATHMathSciNetGoogle Scholar
  14. 14.
    Flum, J., Grohe, M.: Parameterized Complexity Theory. Springer, Berlin (2006) Google Scholar
  15. 15.
    Fowler, R.J., Paterson, M.S., Tanimoto, S.L.: Optimal packing and covering in the plane are NP-complete. Inf. Process. Lett. 12, 133–137 (1981) CrossRefMATHMathSciNetGoogle Scholar
  16. 16.
    Frank, A.: Some polynomial algorithms for certain graphs and hypergraphs. In: Proc. 5th British Combinatorial Conference, Aberdeen, 1975. Congr. Numer, vol. 15, pp. 211–226 (1976) Google Scholar
  17. 17.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1, 237–267 (1976) CrossRefMATHMathSciNetGoogle Scholar
  18. 18.
    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) CrossRefMATHMathSciNetGoogle Scholar
  19. 19.
    Gibson, M., Pirwani, I.A.: Algorithms for dominating set in disk graphs: breaking the logn barrier. In: Proc. 18th Annual European Symposium on Algorithms (ESA 2010). LNCS, vol. 6346, pp. 243–254. Springer, Berlin (2010) Google Scholar
  20. 20.
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980) MATHGoogle Scholar
  21. 21.
    Golumbic, M.C.: Algorithmic aspects of intersection graphs and representation hypergraphs. Graphs Comb. 4, 307–321 (1988) CrossRefMATHMathSciNetGoogle Scholar
  22. 22.
    Gräf, A.: Coloring and recognizing special graph classes. PhD thesis, Technical Report Musikinformatik und Medientechnik Bericht 20/95, Johannes Gutenberg-Universität Mainz (1995) Google Scholar
  23. 23.
    Griggs, J.R., West, D.B.: Extremal values of the interval number of a graph. SIAM J. Algebr. Discrete Methods 1, 1–7 (1980) CrossRefMATHMathSciNetGoogle Scholar
  24. 24.
    Hoefer, M., Kesselheim, T., Vöcking, B.: Approximation algorithms for secondary spectrum auctions. In: Proc. 23rd ACM Symposium on Parallelism in Algorithms and Architectures (SPAA 11), pp. 177–186 (2011) CrossRefGoogle Scholar
  25. 25.
    Hunt, H.B. III, Marathe, M.V., Radhakrishnan, V., Ravi, S.S., Rosenkrantz, D.J., Stearns, R.E.: NC-approximation schemes for NP- and PSPACE-hard problems for geometric graphs. J. Algorithms 26, 238–274 (1998) CrossRefMATHMathSciNetGoogle Scholar
  26. 26.
    Hurink, J.L., Nieberg, T.: Approximating minimum independent dominating sets in wireless networks. Inf. Process. Lett. 109, 155–160 (2008) CrossRefMathSciNetGoogle Scholar
  27. 27.
    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) CrossRefMATHMathSciNetGoogle Scholar
  28. 28.
    Jamison, R.E., Mulder, H.M.: Tolerance intersection graphs on binary trees with constant tolerance 3. Discrete Math. 215, 115–131 (2000) CrossRefMATHMathSciNetGoogle Scholar
  29. 29.
    Jiang, M., Zhang, Y.: Parameterized complexity in multiple-interval graphs: partition, separation, irredundancy. In: Proc. 17th International Computing & Combinatorics Conference (COCOON 2011). LNCS, vol. 6842, pp. 62–73. Springer, Berlin (2011) Google Scholar
  30. 30.
    Kammer, F., Tholey, T., Voepel, H.: Approximation algorithms for intersection graphs. Report 2009-6, Institut für Informatik, Universität Augsburg (2009) Google Scholar
  31. 31.
    Kammer, F., Tholey, T., Voepel, H.: Approximation algorithms for intersection graphs. In: Proc. 13th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX 2010) and 14th International Workshop on Randomization and Computation (RANDOM 2010). LNCS, vol. 6302, pp. 260–273. Springer, Berlin (2010) CrossRefGoogle Scholar
  32. 32.
    Malesińska, E.: Graph-theoretical models for frequency assignment problems. PhD thesis, University of Berlin (1997) Google Scholar
  33. 33.
    Marathe, M.V., Breu, H., Hunt, H.B. III, Ravi, S.S., Rosenkrantz, D.J.: Simple heuristics for unit disk graphs. Networks 25, 59–68 (1995) CrossRefMATHMathSciNetGoogle Scholar
  34. 34.
    Nieberg, T., Hurink, J., Kern, W.: Approximation schemes for wireless networks. ACM Trans. Algorithms 4, Article 49 (2008) CrossRefMathSciNetGoogle Scholar
  35. 35.
    Niedermeier, R.: Invitation to Fixed-Parameter Algorithms. Oxford University Press, New York (2006) CrossRefMATHGoogle Scholar
  36. 36.
    Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. Syst. Sci. 43, 425–440 (1991) CrossRefMATHMathSciNetGoogle Scholar
  37. 37.
    Pirwani, I.A., Salavatipour, M.R.: A weakly robust PTAS for minimum clique partition in unit disk graphs. In: Proc. 12th Scandinavian Symposium and Workshops on Algorithms Theory (SWAT 2010). LNCS, vol. 6139, pp. 188–199. Springer, Berlin (2010). Full version to appear in Algorithmica Google Scholar
  38. 38.
    Rose, D.J., Tarjan, R.E.: Algorithmic aspects of vertex elimination. In: Proc. 7th Annual ACM Symposium on Theory of Computing (STOC 1975), pp. 245–254 (1975) Google Scholar
  39. 39.
    Tardos, É.: A strongly polynomial algorithm to solve combinatorial linear programs. Oper. Res. 34, 250–256 (1986) CrossRefMATHMathSciNetGoogle Scholar
  40. 40.
    Ye, Y., Borodin, A.: Elimination graphs. In: 36th International Colloquium on Automata, Languages and Programming (ICALP 2009). LNCS, vol. 5555, pp. 774–785. Springer, Berlin (2009) CrossRefGoogle Scholar
  41. 41.
    Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. Theory Comput. 3, 103–128 (2007) CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Institut für InformatikAugsburg UniversityAugsburgGermany

Personalised recommendations