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.
This work was partially supported by the Federal Ministry of Education and Research (BMBF) as part of the project ‘Resilience by Spontaneous Volunteers Networks for Coping with Emergencies and Disaster’ (RESIBES), (grant no. 13N13955 to 13N13957).
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
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)
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)
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_21
Awerbuch, B., Azar, Y., Bartal, Y.: Online generalized Steiner problem. Theor. Comput. Sci. 324(2–3), 313–324 (2004)
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)
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_15
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)
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_48
Imase, M., Waxman, B.M.: Dynamic Steiner tree problem. SIAM J. Discrete Math. 4(3), 369–384 (1991)
Klein, P., Ravi, R.: A nearly best-possible approximation algorithm for node-weighted Steiner trees. J. Algorithms 19(1), 104–115 (1995)
Korman, S.: On the use of randomization in the online set cover problem. Master’s thesis, Weizmann Institute of Science, Israel (2005)
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)
(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)
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_4
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)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Markarian, C. (2018). An Optimal Algorithm for Online Prize-Collecting Node-Weighted Steiner Forest. In: Iliopoulos, C., Leong, H., Sung, WK. (eds) Combinatorial Algorithms. IWOCA 2018. Lecture Notes in Computer Science(), vol 10979. Springer, Cham. https://doi.org/10.1007/978-3-319-94667-2_18
Download citation
DOI: https://doi.org/10.1007/978-3-319-94667-2_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-94666-5
Online ISBN: 978-3-319-94667-2
eBook Packages: Computer ScienceComputer Science (R0)