Skip to main content
SpringerLink
Log in

Online Constrained Forest and Prize-Collecting Network Design

  • Published:
Algorithmica Aims and scope Submit manuscript

Abstract

In this paper, we study a very general type of online network design problem, and generalize two different previous algorithms, one for an online network design problem due to Berman and Coulston (Proceedings of the 29th annual ACM symposium on theory of computing, pp 344–353, 1997) and one for (offline) general network design problems due to Goemans and Williamson (SIAM J Comput 24:296–317, 1995); we give an \(O(\log k)\)-competitive algorithm, where k is the number of nodes that must be connected. We also consider a further generalization of the problem that allows us to pay penalties in exchange for violating connectivity constraints; we give an online \(O(\log k)\)-competitive algorithm for this case as well.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

References

  1. Agrawal, A., Klein, P., Ravi, R.: When trees collide: an approximation algorithm for the generalized Steiner problem on networks. SIAM J. Comput. 24, 440–456 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  2. Awerbuch, B., Azar, Y., Bartal, Y.: On-line generalized Steiner problem. Theor. Comput. Sci. 324, 313–324 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  3. Ball, M.O., Magnanti, T.L., Monma, C.L., Nemhauser, G.L. (eds.): Network Models, Handbooks in Operations Research and Management Science, vol. 7. Elsevier, Amsterdam (1995)

    Google Scholar 

  4. Berman, P., Coulston, C.: On-line algorithms for Steiner tree problems. In: Proceedings of the 29th Annual ACM Symposium on Theory of Computing, pp. 344–353 (1997)

  5. Bienstock, D., Goemans, M.X., Simchi-Levi, D., Williamson, D.P.: A note on the prize collecting traveling salesman problem. Math. Program. 59, 413–420 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  6. Cheung, S.S.: Offline and online facility location and network design. Ph.D. thesis, Cornell University, School of Operations Research and Information Engineering (2016)

  7. Goemans, M., Goldberg, A., Plotkin, S., Shmoys, D., Tardos, E., Williamson, D.: Improved approximation algorithms for network design problems. In: Proceedings of the 5th ACM-SIAM Symposium on Discrete Algorithms, pp. 223–232 (1994)

  8. Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. In: Proceedings of the 3rd ACM-SIAM Symposium on Discrete Algorithms, pp. 307–316 (1992)

  9. Goemans, M.X., Williamson, D.P.: A general approximation technique for constrained forest problems. SIAM J. Comput. 24, 296–317 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Gu, A., Gupta, A., Kumar, A.: The power of deferral: maintaining a constant competitive Steiner tree online. In: Proceedings of the 45th Annual ACM Symposium on Theory of Computing, pp. 525–534 (2013)

  11. Gupta, A., Krishnaswamy, R., Ravi, R.: Online and stochastic survivable network design. In: Proceedings of the 41st Annual ACM Symposium on Theory of Computing, pp. 685–694 (2009)

  12. Hajiaghayi, M., Jain, K.: Prize-collecting generalized Steiner tree problem via a new approach of primal–dual schema. In: Proceedings of the 17th ACM-SIAM Symposium on Discrete Algorithms, pp. 631–640 (2006)

  13. Hajiaghayi, M., Liaghat, V., Panigrahi, D.: Online node-weighted Steiner forest and extensions via disk paintings. In: Proceedings of the 54th Annual Symposium on Foundations of Computer Science, pp. 558–567 (2013)

  14. Hajiaghayi, M., Liaghat, V., Panigrahi, D.: Near-optimal online algorithms for prize-collecting Steiner problems. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) Automata, Languages, and Programming, 41st International Colloquium, ICALP 2014, Lecture Notes in Computer Science, vol. 8572, pp. 576–587. Springer (2014)

  15. Hayrapetyan, A., Swamy, C., Tardos, É.: Network design for information networks. In: Proceedings of the 16th ACM-SIAM Symposium on Discrete Algorithms, pp. 933–942 (2005)

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

    Article  MathSciNet  MATH  Google Scholar 

  17. Johnson, D.S., Minkoff, M., Phillips, S.: The prize collecting Steiner tree problem: theory and practice. In: Proceedings of the 11th ACM-SIAM Symposium on Discrete Algorithms, pp. 760–769 (2000)

  18. Karp, R.M.: Reducibility among combinatorial problems. In: Miller, R., Thatcher, J. (eds.) Complexity of Computer Computations, pp. 85–103. Plenum Press, New York (1972)

    Chapter  Google Scholar 

  19. Megow, N., Skutella, M., Verschae, J., Wiese, A.: The power of recourse for online MST and TSP. In: Czumaj, A., Mehlhorn, K., Pitts, A.M., Wattenhofer, R. (eds.) Automata, Languages, and Programming, no. 7391 in Lecture Notes in Computer Science, pp. 689–700. Springer (2012)

  20. Qian, J., Williamson, D.P.: An \({O}(\log n)\)-competitive algorithm for online constrained forest problems. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) Automata, Languages, and Programming, no. 6755 in Lecture Notes in Computer Science, pp. 37–48. Springer, Berlin (2011)

  21. Sharma, Y., Swamy, C., Williamson, D.P.: Approximation algorithms for prize collecting forest problems with submodular penalty functions. In: Proceedings of the 18th ACM-SIAM Symposium on Discrete Algorithms, pp. 1275–1284 (2007)

  22. Umboh, S.: Online network design algorithms via hierarchical decompositions. In: Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 1373–1387 (2015)

  23. Williamson, D.P., Goemans, M.X., Mihail, M., Vazirani, V.V.: A primal–dual approximation algorithm for generalized Steiner network problems. Combinatorica 15, 435–454 (1995)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. JW Capital, Beijing, China

    Jiawei Qian

  2. Department of Mathematics and Computer Science, Eindhoven University of Technology, 5600 MB, Eindhoven, Netherlands

    Seeun William Umboh

  3. School of Operations Research and Information Engineering, Cornell University, Ithaca, NY, 14853, USA

    David P. Williamson

Authors
  1. Jiawei Qian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Seeun William Umboh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. David P. Williamson
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to David P. Williamson.

Additional information

This work was supported in part by the NSF Grant CCF-1115256 and NWO Vidi Grant 639.022.211. Part of this work was done while the second author was visiting the Simons Institute for the Theory of Computing. A preliminary version of this paper appeared in ICALP 2011 [20].

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qian, J., Umboh, S.W. & Williamson, D.P. Online Constrained Forest and Prize-Collecting Network Design. Algorithmica 80, 3335–3364 (2018). https://doi.org/10.1007/s00453-017-0391-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00453-017-0391-4

Keywords

Search

Navigation