Skip to main content
SpringerLink
Log in

Densest k-Subgraph Approximation on Intersection Graphs

  • Conference paper
Approximation and Online Algorithms (WAOA 2010)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 6534))

Included in the following conference series:

Abstract

We study approximation solutions for the densest k-subgraph problem (DS-k) on several classes of intersection graphs. We adopt the concept of σ-quasi elimination orders, introduced by Akcoglu et al. [1], generalizing the perfect elimination orders for chordal graphs, and develop a simple O(σ)-approximation technique for graphs admitting such a vertex order. This concept allows us to derive constant factor approximation algorithms for DS-k on many intersection graph classes, such as chordal graphs, circular-arc graphs, claw-free graphs, line graphs of ℓ-hypergraphs, disk graphs, and the intersection graphs of fat geometric objects. We also present a PTAS for DS-k on unit disk graphs using the shifting technique.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Akcoglu, K., Aspnes, J., DasGupata, B., Kao, M.-Y.: Opportunity cost algorithm for combinatorial auctions. In: Applied Optimization 74: Computational Methods in Decision-Making, Economics and Finance, pp. 455–479 (2002)

    Google Scholar 

  2. Andersen, R., Chellapilla, K.: Finding dense subgraphs with size bounds. In: Avrachenkov, K., Donato, D., Litvak, N. (eds.) WAW 2009. LNCS, vol. 5427, pp. 25–37. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  3. Arora, S., Karger, D., Karpinski, M.: Polynomial time approximation schemes for dense instances of NP-hard problems. In: Proceedings of the 27th Annual ACM Symposium on Theory of Computing (STOC 1995), pp. 284–293 (1995)

    Google Scholar 

  4. Asahiro, Y., Iwama, K., Tamaki, H., Tokuyama, T.: Greedily finding a dense subgraph. In: Karlsson, R., Lingas, A. (eds.) SWAT 1996. LNCS, vol. 1097, pp. 136–148. Springer, Heidelberg (1996)

    Chapter  Google Scholar 

  5. Backer, J., Keil, J.M.: Constant factor approximation algorithms for the densest k-subgraph problem on proper interval graphs and bipartite permutation graphs. Information Processing Letters 110(16), 635–638 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Baker, B.: Approximation algorithms for NP-complete problems on planar graphs. Journal of the ACM 41(1), 153–180 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bhaskara, A., Charikar, M., Chlamtac, E., Feige, U., Vijayaraghavan, A.: Detecting high log-densities — An O(n 1/4) approximation for densest k-subgraph. In: Proceedings of the 42nd Annual ACM Symposium on Theory of Computing (STOC 2010), pp. 201–210 (2010)

    Google Scholar 

  8. Breu, H.: Algorithmic aspects of constrained unit disk graphs. PhD thesis, University of British Columbia (1996)

    Google Scholar 

  9. Breu, H., Kirkpartrick, D.G.: Unit disc graph recognition is NP-hard. Computational Geometry: Theory and Applications 9, 3–24 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  10. Corneil, D.G., Perl, Y.: Cluster and domination in perfect graphs. Discrete Applied Mathematics 9, 27–39 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  11. Demaine, E.D., Hajiaghayi, M., Kawarabayashi, K.: Algorithmic graph minor theory: Decomposition, approximation, and coloring. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), pp. 637–646 (2005)

    Google Scholar 

  12. Efrat, A., Katz, M.J., Nielsenc, F., Sharir, M.: Dynamic data structures for fat objects and their applications. Computational Geometry: Theory and Applications 15, 215–227 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  13. Erlebach, T., Jansen, K., Seidel, E.: Polynomial-time approximation scheme for geometric graphs. In: Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA 2001), pp. 671–679 (2001)

    Google Scholar 

  14. Feige, U., Kortsarz, G., Peleg, D.: The dense k-subgraph problem. Algorithmica 29, 410–421 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  15. Feige, U., Selter, M.: On the densest k-subgraph problem. Technical report, Department of Applied Mathematics and Computer Science, the Weizmann Institute, Rehovot (September 1997)

    Google Scholar 

  16. Gallo, G., Grigoriadis, M.D., Tarjan, R.E.: A fast parametric maximum flow algorithm and applications. SIAM Journal on Computing 18(1), 30–50 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)

    MATH  Google Scholar 

  18. Grötschel, M., Lovász, L., Schrijver, A.: Geometric Algorithms and Combinatorial Optimization. Springer, Heidelberg (1988)

    Book  MATH  Google Scholar 

  19. Han, Q., Ye, Y., Zhang, J.: Approximation of dense-k-subgraph (2000) (manuscript)

    Google Scholar 

  20. Keil, J.M., Brecht, T.B.: The complexity of clustering in planar graphs. The Journal of Combinatorial Mathematics and Combinatorial Computing 9, 155–159 (1991)

    MathSciNet  MATH  Google Scholar 

  21. Khot, S.: Ruling out PTAS for graph min-bisection, densest subgraph and bipartite clique. In: Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2004), pp. 136–145 (2004)

    Google Scholar 

  22. Khuller, S., Saha, B.: On finding dense subgraphs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 597–608. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

  23. Kortsarz, G., Peleg, D.: On choosing a dense subgraph. In: Proceedings of the 37th Annual IEEE Symposium on Foundations of Computer Science (FOCS 1993), pp. 692–701 (1993)

    Google Scholar 

  24. Faudree, R., Flandrin, E., Ryjáek, Z.: Claw-free graphs — A survey. Discrete Mathematics 164(1-3), 87–147 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  25. Liazi, M., Milis, I., Zissimopoulos, V.: Polynomial variants of the densest/heaviest k-subgraph problem. In: Proceedings of the 20th British Combinatorial Conference, Durham (2005)

    Google Scholar 

  26. Liazi, M., Milis, I., Zissimopoulos, V.: The densest k-subgraph problem on clique graphs. Journal of Combinatorial Optimization 14(4), 465–474 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Liazi, M., Milis, I., Zissimopoulos, V.: A constant approximation algorithm for the densest k-subgraph problem on chordal graphs. Information Processing Letters 108(1), 29–32 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Maffioli, F.: Finding a best subtree of a tree. Technical Report 91.041, Politecnico di Milano, Dipartimento di Elettronica, Italy (1991)

    Google Scholar 

  29. Matsui, T.: Approximation algorithms for maximum independent set problems and fractional coloring problems on unit disk graphs. In: Akiyama, J., Kano, M., Urabe, M. (eds.) JCDCG 1998. LNCS, vol. 1763, pp. 194–200. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  30. Peeters, R.: The maximum edge biclique problem is NP-complete. Discrete Applied Mathematics 131(33), 651–654 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  31. Roberts, F.S.: Indifference graphs. In: Harary, F. (ed.) Proof Techniques in Graph Theory. Academic Press, New York (1969)

    Google Scholar 

  32. Srivastav, A., Wolf, K.: Finding dense subgraphs with semidefinite programming. In: Jansen, K., Rolim, J.D.P. (eds.) APPROX 1998. LNCS, vol. 1444, pp. 181–191. Springer, Heidelberg (1998)

    Chapter  Google Scholar 

  33. Turán, P.: On the theory of graphs. Colloquium Mathematicum 3, 19–30 (1954)

    MATH  Google Scholar 

  34. Ye, Y., Zhang, J.: Approximation of dense \(n\over 2\)-subgraph problem and the complement of min-bisection. Journal of Global Optimization 25(1), 55–73 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  35. Ye, Y., Borodin, A.: Sequential elimination graphs. In: Albers, S., Marchetti-Spaccamela, A., Matias, Y., Nikoletseas, S., Thomas, W. (eds.) ICALP 2009. LNCS, vol. 5555, pp. 774–785. Springer, Heidelberg (2009)

    Chapter  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. University of Notre Dame, Indiana, USA

    Danny Z. Chen

  2. SCS and IIPL, Fudan University, Shanghai, China

    Rudolf Fleischer

  3. University of Maryland, College Park, MD, USA

    Jian Li

Authors
  1. Danny Z. Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rudolf Fleischer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jian Li
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Kiel, Olshausenstrasse 40, 24118, Kiel, Germany

    Klaus Jansen

  2. Department of Computer Science, The University of Western Ontario, N6A 5B7, London, Ontario, Canada

    Roberto Solis-Oba

Rights and permissions

Reprints and permissions

Copyright information

© 2011 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Chen, D.Z., Fleischer, R., Li, J. (2011). Densest k-Subgraph Approximation on Intersection Graphs. In: Jansen, K., Solis-Oba, R. (eds) Approximation and Online Algorithms. WAOA 2010. Lecture Notes in Computer Science, vol 6534. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18318-8_8

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-18318-8_8

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-18317-1

  • Online ISBN: 978-3-642-18318-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation