A Nearly Optimal Deterministic Online Algorithm for Non-Metric Facility Location

Authors Marcin Bienkowski , Björn Feldkord , Paweł Schmidt

Thumbnail PDF


  • Filesize: 0.77 MB
  • 17 pages

Document Identifiers

Author Details

Marcin Bienkowski
  • Institute of Computer Science, University of Wrocław, Poland
Björn Feldkord
  • Heinz Nixdorf Institut & Department of Computer Science, Universität Paderborn, Germany
Paweł Schmidt
  • Institute of Computer Science, University of Wrocław, Poland


We thank Marek Adamczyk and Christine Markarian for helpful discussions. We thank anonymous reviewers of an earlier draft for pointing us to the reduction of Kolen and Tamir [Kolen and Tamir, 1990].

Cite AsGet BibTex

Marcin Bienkowski, Björn Feldkord, and Paweł Schmidt. A Nearly Optimal Deterministic Online Algorithm for Non-Metric Facility Location. In 38th International Symposium on Theoretical Aspects of Computer Science (STACS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 187, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


In the online non-metric variant of the facility location problem, there is a given graph consisting of a set F of facilities (each with a certain opening cost), a set C of potential clients, and weighted connections between them. The online part of the input is a sequence of clients from C, and in response to any requested client, an online algorithm may open an additional subset of facilities and must connect the given client to an open facility. We give an online, polynomial-time deterministic algorithm for this problem, with a competitive ratio of O(log |F| ⋅ (log |C| + log log |F|)). The result is optimal up to loglog factors. Our algorithm improves over the O((log |C| + log |F|) ⋅ (log |C| + log log |F|))-competitive construction that first reduces the facility location instance to a set cover one and then later solves such instance using the deterministic algorithm by Alon et al. [TALG 2006]. This is an asymptotic improvement in a typical scenario where |F| ≪ |C|. We achieve this by a more direct approach: we design an algorithm for a fractional relaxation of the non-metric facility location problem with clustered facilities. To handle the constraints of such non-covering LP, we combine the dual fitting and multiplicative weight updates approach. By maintaining certain additional monotonicity properties of the created fractional solution, we can handle the dependencies between facilities and connections in a rounding routine. Our result, combined with the algorithm by Naor et al. [FOCS 2011] yields the first deterministic algorithm for the online node-weighted Steiner tree problem. The resulting competitive ratio is O(log k ⋅ log² 𝓁) on graphs of 𝓁 nodes and k terminals.

Subject Classification

ACM Subject Classification
  • Theory of computation → Online algorithms
  • Theory of computation → Routing and network design problems
  • Online algorithms
  • deterministic rounding
  • linear programming
  • facility location
  • set cover


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Karen Aardal, Jaroslaw Byrka, and Mohammad Mahdian. Facility location. In Encyclopedia of Algorithms, pages 717-724. Springer, 2016. URL: https://doi.org/10.1007/978-1-4939-2864-4_139.
  2. Noga Alon, Baruch Awerbuch, Yossi Azar, Niv Buchbinder, and Joseph Naor. A general approach to online network optimization problems. ACM Transactions on Algorithms, 2(4):640-660, 2006. URL: https://doi.org/10.1145/1198513.1198522.
  3. Noga Alon, Baruch Awerbuch, Yossi Azar, Niv Buchbinder, and Joseph Naor. The online set cover problem. SIAM Journal on Computing, 39(2):361-370, 2009. URL: https://doi.org/10.1137/060661946.
  4. Aris Anagnostopoulos, Russell Bent, Eli Upfal, and Pascal Van Hentenryck. A simple and deterministic competitive algorithm for online facility location. Information and Computation, 194(2):175-202, 2004. URL: https://doi.org/10.1016/j.ic.2004.06.002.
  5. Sanjeev Arora, Elad Hazan, and Satyen Kale. The multiplicative weights update method: a meta-algorithm and applications. Theory of Computing Systems, 8(1):121-164, 2012. URL: https://doi.org/10.4086/toc.2012.v008a006.
  6. Baruch Awerbuch, Yossi Azar, and Serge A. Plotkin. Throughput-competitive on-line routing. In Proc. 34th IEEE Symp. on Foundations of Computer Science (FOCS), pages 32-40, 1993. URL: https://doi.org/10.1109/SFCS.1993.366884.
  7. Niv Buchbinder and Joseph Naor. The design of competitive online algorithms via a primal-dual approach. Foundations and Trends in Theoretical Computer Science, 3(2-3):93-263, 2009. URL: https://doi.org/10.1561/0400000024.
  8. Niv Buchbinder and Joseph Naor. Online primal-dual algorithms for covering and packing. Mathematics of Operations Research, 34(2):270-286, 2009. URL: https://doi.org/10.1287/moor.1080.0363.
  9. Jaroslaw Byrka and Karen Aardal. An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem. SIAM Journal on Computing, 39(6):2212-2231, 2010. URL: https://doi.org/10.1137/070708901.
  10. Moses Charikar and Sudipto Guha. Improved combinatorial algorithms for facility location problems. SIAM Journal on Computing, 34(4):803-824, 2005. URL: https://doi.org/10.1137/S0097539701398594.
  11. Fabián A. Chudak and David B. Shmoys. Improved approximation algorithms for the uncapacitated facility location problem. SIAM Journal on Computing, 33(1):1-25, 2003. URL: https://doi.org/10.1137/S0097539703405754.
  12. Uriel Feige. A threshold of ln n for approximating set cover. Journal of the ACM, 45(4):634-652, 1998. URL: https://doi.org/10.1145/285055.285059.
  13. Dimitris Fotakis. A primal-dual algorithm for online non-uniform facility location. Journal of Discrete Algorithms, 5(1):141-148, 2007. URL: https://doi.org/10.1016/j.jda.2006.03.001.
  14. Dimitris Fotakis. On the competitive ratio for online facility location. Algorithmica, 50(1):1-57, 2008. URL: https://doi.org/10.1007/s00453-007-9049-y.
  15. MohammadTaghi Hajiaghayi, Vahid Liaghat, and Debmalya Panigrahi. Near-optimal online algorithms for prize-collecting steiner problems. In Proc. 41st Int. Colloq. on Automata, Languages and Programming (ICALP), pages 576-587, 2014. URL: https://doi.org/10.1007/978-3-662-43948-7_48.
  16. MohammadTaghi Hajiaghayi, Vahid Liaghat, and Debmalya Panigrahi. Online node-weighted steiner forest and extensions via disk paintings. SIAM Journal on Computing, 46(3):911-935, 2017. URL: https://doi.org/10.1137/14098692X.
  17. Dorit S. Hochbaum. Heuristics for the fixed cost median problem. Mathematical Programming, 22(1):148-162, 1982. URL: https://doi.org/10.1007/BF01581035.
  18. Kamal Jain, Mohammad Mahdian, Evangelos Markakis, Amin Saberi, and Vijay V. Vazirani. Greedy facility location algorithms analyzed using dual fitting with factor-revealing LP. Journal of the ACM, 50(6):795-824, 2003. URL: https://doi.org/10.1145/950620.950621.
  19. Kamal Jain, Mohammad Mahdian, and Amin Saberi. A new greedy approach for facility location problems. In Proc. 34th ACM Symp. on Theory of Computing (STOC), pages 731-740, 2002. URL: https://doi.org/10.1145/509907.510012.
  20. Antoon Kolen and Arie Tamir. Covering problems. In P.B. Mirchandani and R.L. Francis, editors, Discrete Location Theory, Wiley Series in Discrete Mathematics and Optimization. Wiley, 1990. Google Scholar
  21. Simon Korman. On the use of randomization in the online set cover problem. Master’s thesis, The Weizmann Institute of Science, 2004. Google Scholar
  22. Madhukar R. Korupolu, C. Greg Plaxton, and Rajmohan Rajaraman. Analysis of a local search heuristic for facility location problems. Journal of Algorithms, 37(1):146-188, 2000. URL: https://doi.org/10.1006/jagm.2000.1100.
  23. Shi Li. A 1.488 approximation algorithm for the uncapacitated facility location problem. Information and Computation, 222:45-58, 2013. URL: https://doi.org/10.1016/j.ic.2012.01.007.
  24. Mohammad Mahdian, Yinyu Ye, and Jiawei Zhang. Approximation algorithms for metric facility location problems. SIAM Journal on Computing, 36(2):411-432, 2006. URL: https://doi.org/10.1137/S0097539703435716.
  25. Adam Meyerson. Online facility location. In Proc. 42nd IEEE Symp. on Foundations of Computer Science (FOCS), pages 426-431, 2001. URL: https://doi.org/10.1109/SFCS.2001.959917.
  26. Joseph Naor, Debmalya Panigrahi, and Mohit Singh. Online node-weighted steiner tree and related problems. In Proc. 52nd IEEE Symp. on Foundations of Computer Science (FOCS), pages 210-219, 2011. URL: https://doi.org/10.1109/FOCS.2011.65.
  27. Prabhakar Raghavan. Probabilistic construction of deterministic algorithms: Approximating packing integer programs. Journal of Computer and System Sciences, 37(2):130-143, 1988. URL: https://doi.org/10.1016/0022-0000(88)90003-7.
  28. David B. Shmoys. Approximation algorithms for facility location problems. In Proc. 3rd Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM), pages 27-33, 2000. URL: https://doi.org/10.1007/3-540-44436-X_4.
  29. David B. Shmoys, Éva Tardos, and Karen Aardal. Approximation algorithms for facility location problems (extended abstract). In Proc. 29th ACM Symp. on Theory of Computing (STOC), pages 265-274, 1997. URL: https://doi.org/10.1145/258533.258600.
  30. Neal E. Young. Randomized rounding without solving the linear program. In Proc. 6th ACM-SIAM Symp. on Discrete Algorithms (SODA), pages 170-178, 1995. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail