Skip to main content
SpringerLink
Log in

The Maximum Clique Problem in Multiple Interval Graphs (Extended Abstract)

  • Conference paper
Book cover Graph-Theoretic Concepts in Computer Science (WG 2012)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 7551))

Included in the following conference series:

  • 993 Accesses

Abstract

Multiple interval graphs are variants of interval graphs where instead of a single interval, each vertex is assigned a set of intervals on the real line. We study the complexity of the MAXIMUM CLIQUE problem in several classes of multiple interval graphs. The MAXIMUM CLIQUE problem, or the problem of finding the size of the maximum clique, is known to be NP-complete for t-interval graphs when t ≥ 3 and polynomial-time solvable when t = 1. The problem is also known to be NP-complete in t-track graphs when t ≥ 4 and polynomial-time solvable when t ≤ 2. We show that MAXIMUM CLIQUE is already NP-complete for unit 2-interval graphs and unit 3-track graphs. Further, we show that the problem is APX-complete for 2-interval graphs, 3-track graphs, unit 3-interval graphs and unit 4-track graphs. We also introduce two new classes of graphs called t-circular interval graphs and t-circular track graphs and study the complexity of the MAXIMUM CLIQUE problem in them. On the positive side, we present a polynomial time t-approximation algorithm for WEIGHTED MAXIMUM CLIQUE on t-interval graphs, improving earlier work with approximation ratio 4t.

This work was partially supported by the grant ANR-09-JCJC-0041.

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. Alon, N.: Piercing d-intervals. Discrete and Computational Geometry 19, 333–334 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Asinowski, A., Cohen, E., Golumbic, M.C., Limouzy, V., Lipshteyn, M., Stern, M.: Vertex Intersection Graphs of Paths on a Grid. Journal of Graph Algorithms and Applications 16(2), 129–150 (2012)

    Article  MathSciNet  Google Scholar 

  3. Aumann, Y., Lewenstein, M., Melamud, O., Pinter, R.Y., Yakhini, Z.: Dotted interval graphs and high throughput genotyping. In: Proc. of the 16th Annual Symposium on Discrete Algorithms, SODA 2005, pp. 339–348 (2005)

    Google Scholar 

  4. Bar-Yehuda, R., Halldórsson, M.M., Naor, J.S., Shachnai, H., Shapira, I.: Scheduling split intervals. SIAM J. Comput. 36, 1–15 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  5. Berman, P., Fujito, T.: On Approximation Properties of the Independent Set Problem for Degree 3 Graphs. In: Sack, J.-R., Akl, S.G., Dehne, F., Santoro, N. (eds.) WADS 1995. LNCS, vol. 955, pp. 449–460. Springer, Heidelberg (1995)

    Chapter  Google Scholar 

  6. Butman, A., Hermelin, D., Lewenstein, M., Rawitz, D.: Optimization problems in multiple-interval graphs. In: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2007, pp. 268–277 (2007)

    Google Scholar 

  7. Cabello, S., Cardinal, J., Langerman, S.: The Clique Problem in Ray Intersection Graph. arXiv (November 2011), http://arxiv.org/pdf/1111.5986.pdf

  8. Chlebík, M., Chlebíková, J.: The complexity of combinatorial optimization problems on d-dimensional boxes. SIAM Journal on Discrete Mathematics 21(1), 158–169 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  9. Crochemore, M., Hermelin, D., Landau, G.M., Vialette, S.: Approximating the 2-Interval Pattern Problem. In: Brodal, G.S., Leonardi, S. (eds.) ESA 2005. LNCS, vol. 3669, pp. 426–437. Springer, Heidelberg (2005)

    Chapter  Google Scholar 

  10. Garey, M.R., Johnson, D.S.: Rectilinear steiner tree problem is NP-complete. SIAM J. Appl. Math. 6, 826–834 (1977)

    Article  MathSciNet  Google Scholar 

  11. Gavril, F.: Algorithms for a maximum clique and a maximum independent set of a circle graph. Networks 3, 261–273 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  12. Gavril, F.: Maximum weight independent sets and cliques in intersection graphs of filaments. Information Processing Letters 73(56), 181–188 (2000)

    Article  MathSciNet  Google Scholar 

  13. Hochbaum, D.S., Levin, A.: Cyclical scheduling and multi-shift scheduling: Complexity and approximation algorithms. Disc. Optimiz. 3(4), 327–340 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hsu, W.-L.: Maximum weight clique algorithms for circular-arc graphs and circle graphs. SIAM J. Comput. 14(1), 224–231 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  15. Jiang, M.: Clique in 3-track interval graphs is APX-hard. arXiv (April 2012), http://arxiv.org/pdf/1204.2202v1.pdf

  16. Jiang, M., Zhang, Y.: Parameterized Complexity in Multiple-Interval Graphs: Domination, Partition, Separation, Irredundancy. arXiv (October 2011), http://arxiv.org/pdf/1110.0187v1.pdf

  17. Kaiser, T.: Transversals of d-Intervals. Discrete Comput. Geom. 18, 195–203 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. Kammer, F., Tholey, T., Voepel, H.: Approximation Algorithms for Intersection Graphs. In: Serna, M., Shaltiel, R., Jansen, K., Rolim, J. (eds.) APPROX 2010, LNCS, vol. 6302, pp. 260–273. Springer, Heidelberg (2010)

    Chapter  Google Scholar 

  19. König, F.G.: Sorting with objectives. PhD thesis, Technische Universität Berlin (2009)

    Google Scholar 

  20. Kratochvíl, J., Nešetřil, J.: Independent set and clique problems in intersection-defined classes of graphs. Commentationes Mathematicae Universitatis Carolinae 31, 85–93 (1990)

    MathSciNet  MATH  Google Scholar 

  21. Middendorf, M., Pfeiffer, F.: The max clique problem in classes of string-graphs. Discrete Mathematics 108, 365–372 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  22. Papadimitriou, C.H., Yannakakis, M.: Optimization, approximation, and complexity classes. J. Comput. System Sci. 43, 425–440 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  23. Scheinerman, E.R.: The maximum interval number of graphs with given genus. Journal of Graph Theory 11(3), 441–446 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  24. Scheinerman, E.R., West, D.B.: The interval number of a planar graph: Three intervals suffice. Journal of Combinatorial Theory, Series B 35(3), 224–239 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  25. Trotter, W.T., Harary, F.: On double and multiple interval graphs. Journal of Graph Theory 3(3), 205–211 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  26. Valiant, L.G.: Universality considerations in VLSI circuits. IEEE Transactions on Computers 30(2), 135–140 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  27. West, D.B., Shmoys, D.B.: Recognizing graphs with fixed interval number is NP-complete. Discrete Applied Mathematics 8(3), 295–305 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  28. Zuckerman, D.: Linear degree extractors and the inapproximability of max clique and chromatic number. In: Proc. 38th ACM Symp. Theory of Computing, STOC 2006, pp. 681–690 (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. LIRMM, CNRS et Université Montpellier 2, 161 rue Ada, 34392, Montpellier Cedex 05, France

    Mathew C. Francis, Daniel Gonçalves & Pascal Ochem

Authors
  1. Mathew C. Francis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel Gonçalves
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pascal Ochem
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Caesarea Rothschild Institute (CRI), 31905, Haifa, Israel

    Martin Charles Golumbic

  2. The Academic College of Tel Aviv, Unit of Statistics and Probability Studies, 64044, Yaffo, Israel

    Michal Stern

  3. Shenkar College and Caesarea Rothschild Institute (CRI), University of Haifa, 31905, Haifa, Israel

    Avivit Levy

  4. Caesarea Rothschild Institute (CRI), 31905, Haifa, Israel

    Gila Morgenstern

Rights and permissions

Reprints and permissions

Copyright information

© 2012 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Francis, M.C., Gonçalves, D., Ochem, P. (2012). The Maximum Clique Problem in Multiple Interval Graphs (Extended Abstract). In: Golumbic, M.C., Stern, M., Levy, A., Morgenstern, G. (eds) Graph-Theoretic Concepts in Computer Science. WG 2012. Lecture Notes in Computer Science, vol 7551. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34611-8_9

Download citation

  • DOI: https://doi.org/10.1007/978-3-642-34611-8_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-34610-1

  • Online ISBN: 978-3-642-34611-8

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation