Skip to main content

Editing Graphs Into Few Cliques: Complexity, Approximation, and Kernelization Schemes

  • Conference paper
  • First Online:
Algorithms and Data Structures (WADS 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9214))

Included in the following conference series:

Abstract

Given an undirected graph G and a positive integer k, the NP-hard Sparse Split Graph Editing problem asks to transform G into a graph that consists of a clique plus isolated vertices by performing at most k edge insertions and deletions; similarly, the \(P_3\)-Bag Editing problem asks to transform G into a graph which is the union of two possibly overlapping cliques. We give a simple linear-time 3-approximation algorithm for Sparse Split Graph Editing, an improvement over a more involved known factor-3.525 approximation. Further, we show that \(P_3\)-Bag Editing is NP-complete. Finally, we present a kernelization scheme for both problems and additionally for the 2-Cluster Editing problem. This scheme produces for each fixed \(\varepsilon \) in polynomial time a kernel of order \(\varepsilon k\). This is, to the best of our knowledge, the first example of a kernelization scheme that converges to a known lower bound.

F. Hüffner—Supported by DFG project ALEPH (HU 2139/1).

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

Access this chapter

Subscribe and save

Springer+ Basic
EUR 32.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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.

Similar content being viewed by others

References

  1. Abu-Khzam, F.N., Fernau, H.: Kernels: annotated, proper and induced. In: Bodlaender, H.L., Langston, M.A. (eds.) IWPEC 2006. LNCS, vol. 4169, pp. 264–275. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  2. Bessy, S., Fomin, F.V., Gaspers, S., Paul, C., Perez, A., Saurabh, S., Thomassé, S.: Kernels for feedback arc set in tournaments. J. Comput. System Sci. 77(6), 1071–1078 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Borgatti, S.P., Everett, M.G.: Models of core/periphery structures. Soc. Networks 21(4), 375–395 (1999)

    Article  Google Scholar 

  4. Cai, L.: Fixed-parameter tractability of graph modification problems for hereditary properties. Inf. Process. Lett. 58(4), 171–176 (1996)

    Article  MATH  Google Scholar 

  5. Damaschke, P., Mogren, O.: Editing the simplest graphs. In: Pal, S.P., Sadakane, K. (eds.) WALCOM 2014. LNCS, vol. 8344, pp. 249–260. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  6. Damaschke, P., Mogren, O.: Editing simple graphs. J. Graph Algorithms Appl. 18(4), 557–576 (2014). doi:10.7155/jgaa.00337

    Article  MathSciNet  Google Scholar 

  7. Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer (2013)

    Google Scholar 

  8. Feder, T., Hell, P.: On realizations of point determining graphs, and obstructions to full homomorphisms. Discrete Math. 308(9), 1639–1652 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. Fernau, H.: Parameterized algorithmics: A graph-theoretic approach. Wilhelm-Schickard-Institut für Informatik. Universität Tübingen, Habilitationsschrift (2005)

    Google Scholar 

  10. Fomin, F.V., Kratsch, S., Pilipczuk, M., Pilipczuk, M., Villanger, Y.: Tight bounds for parameterized complexity of cluster editing with a small number of clusters. J. Comput. System Sci. 80(7), 1430–1447 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  11. Garey, M.R., Johnson, D.S., Stockmeyer, L.J.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  12. Giotis, I., Guruswami, V.: Correlation clustering with a fixed number of clusters. Theory of Computing 2(1), 249–266 (2006)

    Article  MathSciNet  Google Scholar 

  13. Guo, J.: A more effective linear kernelization for cluster editing. Theor. Comput. Sci. 410(8–10), 718–726 (2009)

    Article  MATH  Google Scholar 

  14. Hammer, P.L., Simeone, B.: The splittance of a graph. Combinatorica 1(3), 275–284 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hell, P.: Graph partitions with prescribed patterns. Eur. J. Combin. 35, 335–353 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  16. Komusiewicz, C., Uhlmann, J.: Cluster editing with locally bounded modifications. Discrete Appl. Math. 160(15), 2259–2270 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Kováč, I., Selečéniová, I., Steinová, M.: On the clique editing problem. In: Csuhaj-Varjú, E., Dietzfelbinger, M., Ésik, Z. (eds.) MFCS 2014, Part II. LNCS, vol. 8635, pp. 469–480. Springer, Heidelberg (2014)

    Google Scholar 

  18. Lewis, J.M., Yannakakis, M.: The node-deletion problem for hereditary properties is NP-complete. J. Comput. System Sci. 20(2), 219–230 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  19. Natanzon, A., Shamir, R., Sharan, R.: Complexity classification of some edge modification problems. Discrete Appl. Math. 113, 109–128 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Shamir, R., Sharan, R., Tsur, D.: Cluster graph modification problems. Discrete Appl. Math. 144(1–2), 173–182 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Wu, B.Y., Chen, L.-H.: Parameterized algorithms for the 2-clustering problem with minimum sum and minimum sum of squares objective functions. Algorithmica (2014, to appear). doi:10.1007/s00453-014-9874-8

    Google Scholar 

  22. Xie, W.: Obstructions to trigraph homomorphisms. Master’s thesis, School of Computing Science. Simon Fraser University, British Columbia, Canada (2006)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to André Nichterlein .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Hüffner, F., Komusiewicz, C., Nichterlein, A. (2015). Editing Graphs Into Few Cliques: Complexity, Approximation, and Kernelization Schemes. In: Dehne, F., Sack, JR., Stege, U. (eds) Algorithms and Data Structures. WADS 2015. Lecture Notes in Computer Science(), vol 9214. Springer, Cham. https://doi.org/10.1007/978-3-319-21840-3_34

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-21840-3_34

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-21839-7

  • Online ISBN: 978-3-319-21840-3

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics