Skip to main content

A Randomized \(\mathrm {O}(\log n)\)-Competitive Algorithm for the Online Connected Facility Location Problem

Algorithmica volume 76, pages 1139–1157 (2016)Cite this article

Abstract

The Connected Facility Location (CFL) is a network design problem that arises from a combination of the Uncapacitated Facility Location (FL) and the Steiner Tree (ST) problems. The Online Connected Facility Location problem (OCFL) is an online version of the CFL. San Felice et al. (2014) presented a randomized algorithm for the OCFL and proved that it is \(\mathrm {O}(\log ^2 n)\)-competitive, where n is the number of clients. That algorithm combines the sample-and-augment framework of Gupta, Kumar, Pál, and Roughgarden with previous algorithms for the Online Facility Location (OFL) and the Online Steiner Tree (OST) problems. In this paper we use a more precise analysis to show that the same algorithm is \(\mathrm {O}(\log n)\)-competitive. Since there is a lower bound of \(\mathrm {\Omega }(\log n)\) for this problem, our result achieves the best possible competitive ratio, asymptotically.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

References

  1. Bartal, Y., Fiat, A., Rabani, Y.: : Competitive Algorithms for Distributed Data Management. J. Comput. Syst. Sci. 51(3), 341–358. doi:10.1006/jcss.1995.1073

  2. Borodin, A., El-Yaniv, R.: Online Computation and Competitive Analysis. Cambridge University Press, New York, NY (1998)

    MATH  Google Scholar 

  3. Buchbinder, N., Naor, J.S.: The design of competitive online algorithms via a primal-dual approach. Found. Trends Theor. Comput. Sci. 3, 93–263 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. Byrka, J., Aardal, K.: An optimal bifactor approximation algorithm for the metric facility location problem. SIAM J. Comput. 39, 2212–2231 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Eisenbrand, F., Grandoni, F., Rothvoß, T., Schäfer, G.: Connected facility location via random facility sampling and core detouring. J. Comput. Syst. Sci. 76(8), 709–726 (2010). http://linkinghub.elsevier.com/retrieve/

  6. Fotakis, D.: A primal-dual algorithm for online non-uniform facility location. J. Discrete Algorithms 5(1), 141–148 (2007). doi:10.1016/j.jda.2006.03.001

    Article  MathSciNet  MATH  Google Scholar 

  7. Fotakis, D.: On the competitive ratio for online facility location. Algorithmica 50, 1–57 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  8. Fotakis, D.: Online and incremental algorithms for facility location. SIGACT News 42(1), 97–131 (2011). doi:10.1145/1959045.1959065

    Article  Google Scholar 

  9. Gupta, A., Kumar, A., Pál, M., Roughgarden, T.: Approximation via cost sharing: Simpler and better approximation algorithms for network design. J. ACM 54(3) (2007). http://portal.acm.org/citation.cfm?doid=1236457.1236458. Article 11

  10. Gupta, A., Srinivasan, A., Tardos, É.: Cost-sharing mechanisms for network design. Algorithmica 50, 98–119 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hasan, M.K., Jung, H., Chwa, K.Y.: Approximation algorithms for connected facility location. J. Comb. Optim. 16, 155–172 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Imase, M., Waxman, B.M.: Dynamic Steiner tree problem. SIAM J. Discrete Math. 4(3), 369–384 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  13. Jung, H., Hasan, M.K., Chwa, K.Y.: A 6.55 factor primal-dual approximation algorithm for the connected facility location problem. J. Comb. Optim. 18, 258–271 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  14. Li, S.: A 1.488 approximation algorithm for the uncapacitated facility location problem. Inf. Comput. 222, 45–58 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  15. Mahdian, M., Ye, Y., Zhang, J.: Approximation algorithms for metric facility location problems. SIAM J. Comput. 36, 411–432 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Meyerson, A.: Online facility location. In: Foundations of Computer Science, 42nd IEEE Symposium, FOCS 2001, pp. 426–431 (2001). http://dl.acm.org/citation.cfm?id=874063.875567

  17. Nagarajan, C., Williamson, D.P.: Offline and online facility leasing. Discrete Optim. 10(4), 361–370 (2013). doi:10.1016/j.disopt.2013.10.001. http://www.sciencedirect.com/science/article/

  18. San Felice: M.C., Williamson, D.P., Lee, O.: The online connected facility location problem. In: Latin American Theoretical INformatics, 11th Symposium, LATIN 2014, no. 8392 in Lecture Notes in Computer Science, pp. 574–585. Springer, Berlin, Heidelberg (2014)

  19. Shmoys, D.B.: Approximation algorithms for facility location problems. In: Approximation Algorithms for Combinatorial Optimization, 3rd International Workshop, APPROX 2000, no. 1913 in Lecture Notes in Computer Science, pp. 27–33. Springer, Berlin, Germany (2000). http://dl.acm.org/citation.cfm?id=646688.703104

  20. Swamy, C., Kumar, A.: Primal-dual algorithms for connected facility location problems. Algorithmica 40(4), 245–269 (2004)

  21. Umboh, S.: Online network design algorithms via hierarchical decompositions. In: Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA). Society for Industrial and Applied Mathematics, Philadelphia, PA, USA (2015). To appear. CoRR, abs/1410.4240

  22. Vazirani, V.: Approximation Algorithms. Springer-Verlag Berlin Heidelberg, Germany (2003)

    Book  Google Scholar 

  23. Williamson, D.P., Shmoys, D.B.: The Design of Approximation Algorithms. Cambridge University Press, New York, NY (2011)

    Book  MATH  Google Scholar 

Download references

Acknowledgments

We would like to thank two anonymous referees whose suggestions and remarks greatly improved the presentation of this paper.

Author information

Authors and Affiliations

  1. Institute of Computing, Unicamp, Campinas, SP, 13083-852, Brazil

    Mário César San Felice & Orlando Lee

  2. School of Operations Research and Information Engineering, Cornell University, Ithaca, NY, 14853-3801, USA

    David P. Williamson

Authors
  1. Mário César San Felice
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. David P. Williamson
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Orlando Lee
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mário César San Felice.

Additional information

First author supported by Grant No. 2009/15535-1, São Paulo Research Foundation (FAPESP).

Second author supported in part by NSF Grant CCF-1115256.

Third author supported in part by Bolsa de Produtividade do CNPq Proc. 303947/2008-0 and Edital Universal CNPq 477692/2012-5.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

San Felice, M.C., Williamson, D.P. & Lee, O. A Randomized \(\mathrm {O}(\log n)\)-Competitive Algorithm for the Online Connected Facility Location Problem. Algorithmica 76, 1139–1157 (2016). https://doi.org/10.1007/s00453-016-0115-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-016-0115-1

Keywords

  • Online algorithms
  • Competitive analysis
  • Connected facility location
  • Steiner tree
  • Approximation algorithms
  • Randomized algorithms