Advertisement

An Optimal Algorithm for Online Prize-Collecting Node-Weighted Steiner Forest

  • Christine MarkarianEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10979)

Abstract

We study the Online Prize-collecting Node-weighted Steiner Forest problem (OPC-NWSF) in which we are given an undirected graph \(G=(V, E)\) with \(|V| = n\) and node-weight function \(w: V \rightarrow \mathcal {R}^+\). A sequence of k pairs of nodes of G, each associated with a penalty, arrives online. OPC-NWSF asks to construct a subgraph H such that each pair \(\{s, t\}\) is either connected (there is a path between s and t in H) or its associated penalty is paid. The goal is to minimize the weight of H and the total penalties paid. The current best result for OPC-NWSF is a randomized \(\mathcal {O}(\log ^4 n)\)-competitive algorithm due to Hajiaghayi et al. (ICALP 2014). We improve this by proposing a randomized \(\mathcal {O}(\log n \log k)\)-competitive algorithm for OPC-NWSF, which is optimal up to constant factor since OPC-NWSF has a randomized lower bound of \(\varOmega (\log ^2 n)\) due to Korman [11]. Moreover, our result also implies an improvement for two special cases of OPC-NWSF, the Online Prize-collecting Node-weighted Steiner Tree problem (OPC-NWST) and the Online Node-weighted Steiner Forest problem (ONWSF). In OPC-NWST, there is a distinguished node which is one of the nodes in each pair. In ONWSF, all penalties are set to infinity. The currently best known results for OPC-NWST and ONWSF are a randomized \(\mathcal {O}(\log ^3 n)\)-competitive algorithm due to Hajiaghayi et al. (ICALP 2014) and a randomized \(\mathcal {O}(\log n \log ^2 k)\)-competitive algorithm due to Hajiaghayi et al. (FOCS 2013), respectively.

Keywords

Online algorithms Competitive analysis Steiner forest Steiner tree Prize-collecting Node-weighted graphs Penalties 

References

  1. 1.
    Alon, N., Awerbuch, B., Azar, Y.: The online set cover problem. In: Proceedings of the Thirty-Fifth Annual ACM Symposium on Theory of Computing, STOC 2003, pp. 100–105. ACM, New York (2003)Google Scholar
  2. 2.
    Alon, N., Awerbuch, B., Azar, Y., Buchbinder, N., (Seffi) Naor, J.: A general approach to online network optimization problems. ACM Trans. Algorithms 2(4), 640–660 (2006)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Angelopoulos, S.: The node-weighted steiner problem in graphs of restricted node weights. In: Arge, L., Freivalds, R. (eds.) SWAT 2006. LNCS, vol. 4059, pp. 208–219. Springer, Heidelberg (2006).  https://doi.org/10.1007/11785293_21CrossRefGoogle Scholar
  4. 4.
    Awerbuch, B., Azar, Y., Bartal, Y.: Online generalized Steiner problem. Theor. Comput. Sci. 324(2–3), 313–324 (2004)CrossRefGoogle Scholar
  5. 5.
    Berman, P., Coulston, C.: Online algorithms for Steiner tree problems (extended abstract). In: Proceedings of the Twenty-Ninth Annual ACM Symposium on the Theory of Computing, El Paso, Texas, USA, 4–6 May 1997, pp. 344–353 (1997)Google Scholar
  6. 6.
    Bienkowski, M., Kraska, A., Schmidt, P.: A deterministic algorithm for online steiner tree leasing. Algorithms and Data Structures. LNCS, vol. 10389, pp. 169–180. Springer, Cham (2017).  https://doi.org/10.1007/978-3-319-62127-2_15CrossRefGoogle Scholar
  7. 7.
    Hajiaghayi, M.T., Liaghat, V., Panigrahi, D.: Online node-weighted Steiner forest and extensions via disk paintings. In: 54th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2013, Berkeley, CA, USA, 26–29 October 2013, pp. 558–567 (2013)Google Scholar
  8. 8.
    Hajiaghayi, M.T., Liaghat, V., Panigrahi, D.: Near-optimal online algorithms for prize-collecting Steiner problems. In: Esparza, J., Fraigniaud, P., Husfeldt, T., Koutsoupias, E. (eds.) ICALP 2014. LNCS, vol. 8572, pp. 576–587. Springer, Heidelberg (2014).  https://doi.org/10.1007/978-3-662-43948-7_48CrossRefGoogle Scholar
  9. 9.
    Imase, M., Waxman, B.M.: Dynamic Steiner tree problem. SIAM J. Discrete Math. 4(3), 369–384 (1991)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Klein, P., Ravi, R.: A nearly best-possible approximation algorithm for node-weighted Steiner trees. J. Algorithms 19(1), 104–115 (1995)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Korman, S.: On the use of randomization in the online set cover problem. Master’s thesis, Weizmann Institute of Science, Israel (2005)Google Scholar
  12. 12.
    Meyerson, A.: The parking permit problem. In: Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science (FOCS 2005), Pittsburgh, PA, USA, 23–25 October 2005, pp. 274–284 (2005)Google Scholar
  13. 13.
    (Seffi) Naor, J., Panigrahi, D., Singh, M.: Online node-weighted Steiner tree and related problems. In: Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, Washington, DC, USA. IEEE Computer Society, pp. 210–219 (2011)Google Scholar
  14. 14.
    Qian, J., Williamson, D.P.: An O(logn)-competitive algorithm for online constrained forest problems. In: Aceto, L., Henzinger, M., Sgall, J. (eds.) ICALP 2011. LNCS, vol. 6755, pp. 37–48. Springer, Heidelberg (2011).  https://doi.org/10.1007/978-3-642-22006-7_4CrossRefGoogle Scholar
  15. 15.
    Schroeder, J., Guedes, A., Duarte Jr., E.P.: Computing the minimum cut and maximum flow of undirected graphs. Technical report, Department of Informatics, Federal University of Paraná (2004)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Heinz Nixdorf Institute, Computer Science DepartmentPaderborn UniversityPaderbornGermany

Personalised recommendations