Skip to main content
SpringerLink
Log in

Adding Isolated Vertices Makes Some Online Algorithms Optimal

  • Conference paper
  • First Online:
Combinatorial Algorithms (IWOCA 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9538))

Included in the following conference series:

  • 567 Accesses

Abstract

An unexpected difference between online and offline algorithms is observed. The natural greedy algorithms are shown to be worst case online optimal for Online Independent Set and Online Vertex Cover on graphs with “enough” isolated vertices, Freckle Graphs. For Online Dominating Set, the greedy algorithm is shown to be worst case online optimal on graphs with at least one isolated vertex. These algorithms are not online optimal in general. The online optimality results for these greedy algorithms imply optimality according to various worst case performance measures, such as the competitive ratio. It is also shown that, despite this worst case optimality, there are Freckle graphs where the greedy independent set algorithm is objectively less good than another algorithm.

It is shown that it is NP-hard to determine any of the following for a given graph: the online independence number, the online vertex cover number, and the online domination number.

Supported in part by the Villum Foundation, the Stibo-Foundation, and the Danish Council for Independent Research, Natural Sciences.

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 EPUB and 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

References

  1. Allan, R.B., Laskar, R.: On domination and independent domination numbers of a graph. Discrete Math. 23(2), 73–76 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  2. Angelopoulos, S., Dorrigiv, R., López-Ortiz, A.: On the separation and equivalence of paging strategies. In: 18th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 229–237 (2007)

    Google Scholar 

  3. Boyar, J., Favrholdt, L.M.: The relative worst order ratio for online algorithms. ACM Trans. Algorithm 3(2), 22 (2007)

    Article  MathSciNet  Google Scholar 

  4. Boyar, J., Favrholdt, L.M., Medvedev, P.: The relative worst order ratio of online bipartite graph coloring, Unpublished manuscript

    Google Scholar 

  5. Garey, M.R., Johnson, D.S.: Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman & Co., New York (1979). Under Dominating Set on p. 190

    Google Scholar 

  6. Gyárfás, A., Király, Z., Lehel, J.: On-line competitive coloring algorithms. Tech. rep. TR-9703-1, Institute of Mathematics at Eötvös Loránd University (1997). http://www.cs.elte.hu/tr97/

  7. Gyárfás, A., Király, Z., Lehel, J.: On-line 3-chromatic graphs I. Triangle-free graphs. SIAM J. Discrete Math. 12, 385–411 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gyárfás, A., Lehel, J.: First fit and on-line chromatic number of families of graphs. ARS Combinatoria 29C, 168–176 (1990)

    MATH  Google Scholar 

  9. Halldórsson, M.M.: Online coloring known graphs. In: 10th ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 917–918 (1999)

    Google Scholar 

  10. Halldórsson, M.M., Iwama, K., Miyazaki, S., Taketomi, S.: Online independent sets. Theoret. Comput. Sci. 289(2), 953–962 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  11. Kudahl, C.: Deciding the on-line chromatic number of a graph with pre-coloring is PSPACE-complete. In: Paschos, V.T., Widmayer, P. (eds.) CIAC 2015. LNCS, vol. 9079, pp. 313–324. Springer, Heidelberg (2015)

    Chapter  Google Scholar 

  12. Lehel, A.G.J., Kiraly, Z.: On-line graph coloring and finite basis problems. Combinatorics: Paul Erdos is Eighty 1, 207–214 (1993)

    MathSciNet  Google Scholar 

  13. Lovász, L., Saks, M., Trotter, W.: An on-line graph coloring algorithm with sublinear performance ratio. Discrete Math. 75(13), 319–325 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  14. Sleator, D.D., Tarjan, R.E.: Amortized efficiency of list update and paging rules. Commun. ACM 28(2), 202–208 (1985)

    Article  MathSciNet  Google Scholar 

  15. Vishwanathan, S.: Randomized online graph coloring. J. Algorithms 13(4), 657–669 (1992)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors would like to thank Lene Monrad Favrholdt for interesting and helpful discussions.

Author information

Authors and Affiliations

  1. Department of Mathematics and Computer Science, University of Southern Denmark, Odense, Denmark

    Joan Boyar & Christian Kudahl

Authors
  1. Joan Boyar
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Christian Kudahl
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Joan Boyar .

Editor information

Editors and Affiliations

  1. Dept. of Computer Science, University of Verona, Verona, Italy

    Zsuzsanna Lipták

  2. Dept. of Computing and Software, McMaster University, Hamilton, Canada

    William F. Smyth

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Boyar, J., Kudahl, C. (2016). Adding Isolated Vertices Makes Some Online Algorithms Optimal. In: Lipták, Z., Smyth, W. (eds) Combinatorial Algorithms. IWOCA 2015. Lecture Notes in Computer Science(), vol 9538. Springer, Cham. https://doi.org/10.1007/978-3-319-29516-9_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-29516-9_6

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-29515-2

  • Online ISBN: 978-3-319-29516-9

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation