An O(loglog n)-Approximation for Submodular Facility Location

Authors Fateme Abbasi , Marek Adamczyk , Miguel Bosch-Calvo , Jarosław Byrka , Fabrizio Grandoni , Krzysztof Sornat , Antoine Tinguely



PDF
Thumbnail PDF

File

LIPIcs.ICALP.2024.5.pdf
  • Filesize: 0.86 MB
  • 20 pages

Document Identifiers

Author Details

Fateme Abbasi
  • University of Wrocław, Poland
Marek Adamczyk
  • University of Wrocław, Poland
Miguel Bosch-Calvo
  • IDSIA, USI-SUPSI, Lugano, Switzerland
Jarosław Byrka
  • University of Wrocław, Poland
Fabrizio Grandoni
  • IDSIA, USI-SUPSI, Lugano, Switzerland
Krzysztof Sornat
  • AGH University, Kraków, Poland
Antoine Tinguely
  • IDSIA, USI-SUPSI, Lugano, Switzerland

Acknowledgements

We would like to thank the anonymous reviewers for their helpful comments, in particular for pointing out to the simpler and stronger lower bound construction by Gupta [Shalmoli Gupta, 2018]. We would also like to thank Neil Olver for inspiring discussions about applications of their technique in [Thomas Bosman and Neil Olver, 2020] to various covering problems over time.

Cite AsGet BibTex

Fateme Abbasi, Marek Adamczyk, Miguel Bosch-Calvo, Jarosław Byrka, Fabrizio Grandoni, Krzysztof Sornat, and Antoine Tinguely. An O(loglog n)-Approximation for Submodular Facility Location. In 51st International Colloquium on Automata, Languages, and Programming (ICALP 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 297, pp. 5:1-5:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ICALP.2024.5

Abstract

In the Submodular Facility Location problem (SFL) we are given a collection of n clients and m facilities in a metric space. A feasible solution consists of an assignment of each client to some facility. For each client, one has to pay the distance to the associated facility. Furthermore, for each facility f to which we assign the subset of clients S^f, one has to pay the opening cost g(S^f), where g() is a monotone submodular function with g(emptyset)=0. SFL is APX-hard since it includes the classical (metric uncapacitated) Facility Location problem (with uniform facility costs) as a special case. Svitkina and Tardos [SODA'06] gave the current-best O(log n) approximation algorithm for SFL. The same authors pose the open problem whether SFL admits a constant approximation and provide such an approximation for a very restricted special case of the problem. We make some progress towards the solution of the above open problem by presenting an O(loglog n) approximation. Our approach is rather flexible and can be easily extended to generalizations and variants of SFL. In more detail, we achieve the same approximation factor for the natural generalizations of SFL where the opening cost of each facility f is of the form p_f + g(S^f) or w_f * g(S^f), where p_f, w_f >= 0 are input values. We also obtain an improved approximation algorithm for the related Universal Stochastic Facility Location problem. In this problem one is given a classical (metric) facility location instance and has to a priori assign each client to some facility. Then a subset of active clients is sampled from some given distribution, and one has to pay (a posteriori) only the connection and opening costs induced by the active clients. The expected opening cost of each facility f can be modelled with a submodular function of the set of clients assigned to f.

Subject Classification

ACM Subject Classification
  • Theory of computation → Approximation algorithms analysis
  • Theory of computation → Facility location and clustering
  • Theory of computation → Rounding techniques
  • Theory of computation → Online algorithms
Keywords
  • approximation algorithms
  • facility location
  • submodular facility location
  • universal stochastic facility location

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads

References

  1. Fateme Abbasi, Marek Adamczyk, Miguel Bosch-Calvo, Jarosław Byrka, Fabrizio Grandoni, Krzysztof Sornat, and Antoine Tinguely. An O(log log n)-approximation for submodular facility location. CoRR, abs/2211.05474, 2022. URL: https://doi.org/10.48550/arXiv.2211.05474.
  2. Marek Adamczyk, Fabrizio Grandoni, Stefano Leonardi, and Michal Włodarczyk. When the optimum is also blind: A new perspective on universal optimization. In 44th International Colloquium on Automata, Languages, and Programming, ICALP 2017, pages 35:1-35:15, 2017. URL: https://doi.org/10.4230/LIPIcs.ICALP.2017.35.
  3. Sara Ahmadian, Ashkan Norouzi-Fard, Ola Svensson, and Justin Ward. Better guarantees for k-means and Euclidean k-median by primal-dual algorithms. SIAM J. Comput., 49(4), 2020. URL: https://doi.org/10.1137/18M1171321.
  4. Hyung-Chan An, Mohit Singh, and Ola Svensson. LP-based algorithms for capacitated facility location. SIAM J. Comput., 46(1):272-306, 2017. URL: https://doi.org/10.1137/151002320.
  5. Thomas Bosman and Neil Olver. Improved approximation algorithms for inventory problems. In Integer Programming and Combinatorial Optimization - 21st International Conference, IPCO 2020, pages 91-103, 2020. URL: https://doi.org/10.1007/978-3-030-45771-6_8.
  6. Jarosław Byrka and Karen Aardal. An optimal bifactor approximation algorithm for the metric uncapacitated facility location problem. SIAM J. Comput., 39(6):2212-2231, 2010. URL: https://doi.org/10.1137/070708901.
  7. Jarosław Byrka, Thomas W. Pensyl, Bartosz Rybicki, Aravind Srinivasan, and Khoa Trinh. An improved approximation for k-median and positive correlation in budgeted optimization. ACM Trans. Algorithms, 13(2):23:1-23:31, 2017. URL: https://doi.org/10.1145/2981561.
  8. Moses Charikar and Sudipto Guha. Improved combinatorial algorithms for facility location problems. SIAM J. Comput., 34(4):803-824, 2005. URL: https://doi.org/10.1137/S0097539701398594.
  9. Chandra Chekuri and Alina Ene. Approximation algorithms for submodular multiway partition. In IEEE 52nd Annual Symposium on Foundations of Computer Science, FOCS 2011, pages 807-816, 2011. URL: https://doi.org/10.1109/FOCS.2011.34.
  10. Chandra Chekuri and Alina Ene. Submodular cost allocation problem and applications. In Automata, Languages and Programming - 38th International Colloquium, ICALP 2011, pages 354-366, 2011. URL: https://doi.org/10.1007/978-3-642-22006-7_30.
  11. Vincent Cohen-Addad, Hossein Esfandiari, Vahab S. Mirrokni, and Shyam Narayanan. Improved approximations for Euclidean k-means and k-median, via nested quasi-independent sets. In 54th Annual ACM SIGACT Symposium on Theory of Computing, STOC 2022, pages 1621-1628, 2022. URL: https://doi.org/10.1145/3519935.3520011.
  12. Vincent Cohen-Addad, Fabrizio Grandoni, Euiwoong Lee, and Chris Schwiegelshohn. Breaching the 2 LMP approximation barrier for facility location with applications to k-median. In 34th ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, pages 940-986, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch37.
  13. Alina Ene, Jan Vondrák, and Yi Wu. Local distribution and the symmetry gap: Approximability of multiway partitioning problems. In 24th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2013, pages 306-325, 2013. URL: https://doi.org/10.1137/1.9781611973105.23.
  14. Jittat Fakcharoenphol, Satish Rao, and Kunal Talwar. A tight bound on approximating arbitrary metrics by tree metrics. J. Comput. Syst. Sci., 69(3):485-497, 2004. URL: https://doi.org/10.1016/J.JCSS.2004.04.011.
  15. Fedor V. Fomin, Dieter Kratsch, and Gerhard J. Woeginger. Exact (exponential) algorithms for the dominating set problem. In Graph-Theoretic Concepts in Computer Science, 30th International Workshop, WG 2004, pages 245-256, 2004. URL: https://doi.org/10.1007/978-3-540-30559-0_21.
  16. Satoru Fujishige. Submodular functions and optimization, volume 58 of Annals of Discrete Mathematics. Elsevier, 2nd edition, 2005. Google Scholar
  17. Naveen Garg, Anupam Gupta, Stefano Leonardi, and Piotr Sankowski. Stochastic analyses for online combinatorial optimization problems. In 19th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, pages 942-951, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347185.
  18. Kishen N. Gowda, Thomas W. Pensyl, Aravind Srinivasan, and Khoa Trinh. Improved bi-point rounding algorithms and a golden barrier for k-median. In 34th ACM-SIAM Symposium on Discrete Algorithms, SODA 2023, pages 987-1011, 2023. URL: https://doi.org/10.1137/1.9781611977554.ch38.
  19. Fabrizio Grandoni, Anupam Gupta, Stefano Leonardi, Pauli Miettinen, Piotr Sankowski, and Mohit Singh. Set covering with our eyes closed. SIAM J. Comput., 42(3):808-830, 2013. URL: https://doi.org/10.1137/100802888.
  20. Fabrizio Grandoni, Rafail Ostrovsky, Yuval Rabani, Leonard J. Schulman, and Rakesh Venkat. A refined approximation for Euclidean k-means. Inf. Process. Lett., 176:106251, 2022. URL: https://doi.org/10.1016/j.ipl.2022.106251.
  21. Sudipto Guha and Samir Khuller. Greedy strikes back: Improved facility location algorithms. J. Algorithms, 31(1):228-248, 1999. URL: https://doi.org/10.1006/JAGM.1998.0993.
  22. Anupam Gupta, Martin Pál, R. Ravi, and Amitabh Sinha. Sampling and cost-sharing: Approximation algorithms for stochastic optimization problems. SIAM J. Comput., 40(5):1361-1401, 2011. URL: https://doi.org/10.1137/080732250.
  23. Shalmoli Gupta. Approximation algorithms for clustering and facility location problems. PhD thesis, University of Illinois Urbana-Champaign, USA, 2018. URL: https://hdl.handle.net/2142/102419.
  24. Nicole Immorlica, David R. Karger, Maria Minkoff, and Vahab S. Mirrokni. On the costs and benefits of procrastination: Approximation algorithms for stochastic combinatorial optimization problems. In 15th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, pages 691-700, 2004. URL: http://dl.acm.org/citation.cfm?id=982792.982898.
  25. Satoru Iwata, Lisa Fleischer, and Satoru Fujishige. A combinatorial strongly polynomial algorithm for minimizing submodular functions. Journal of the ACM (JACM), 48(4):761-777, 2001. URL: https://doi.org/10.1145/502090.502096.
  26. 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 (JACM), 50(6):795-824, 2003. URL: https://doi.org/10.1145/950620.950621.
  27. Kamal Jain and Vijay V. Vazirani. Approximation algorithms for metric facility location and k-median problems using the primal-dual schema and Lagrangian relaxation. Journal of the ACM (JACM), 48(2):274-296, 2001. URL: https://doi.org/10.1145/375827.375845.
  28. Shi Li. A 1.488 approximation algorithm for the uncapacitated facility location problem. Inf. Comput., 222:45-58, 2013. URL: https://doi.org/10.1016/J.IC.2012.01.007.
  29. Shi Li and Ola Svensson. Approximating k-median via pseudo-approximation. SIAM J. Comput., 45(2):530-547, 2016. URL: https://doi.org/10.1137/130938645.
  30. Mohammad Mahdian, Yinyu Ye, and Jiawei Zhang. Approximation algorithms for metric facility location problems. SIAM J. Comput., 36(2):411-432, 2006. URL: https://doi.org/10.1137/S0097539703435716.
  31. Adam Meyerson. Online facility location. In 42nd Annual Symposium on Foundations of Computer Science, FOCS 2001, pages 426-431, 2001. URL: https://doi.org/10.1109/SFCS.2001.959917.
  32. Alexander Schrijver. Theory of linear and integer programming. John Wiley & Sons, 1999. Google Scholar
  33. David B. Shmoys, Chaitanya Swamy, and Retsef Levi. Facility location with service installation costs. In 15th Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2004, pages 1088-1097, 2004. URL: http://dl.acm.org/citation.cfm?id=982792.982953.
  34. David B. Shmoys, Éva Tardos, and Karen Aardal. Approximation algorithms for facility location problems (extended abstract). In 29th Annual ACM Symposium on the Theory of Computing, STOC 1997, pages 265-274, 1997. URL: https://doi.org/10.1145/258533.258600.
  35. Zoya Svitkina and Éva Tardos. Facility location with hierarchical facility costs. In Proceedings of the Seventeenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2006, pages 153-161, 2006. URL: http://dl.acm.org/citation.cfm?id=1109557.1109576.
  36. Zoya Svitkina and Éva Tardos. Facility location with hierarchical facility costs. ACM Trans. Algorithms, 6(2):37:1-37:22, 2010. URL: https://doi.org/10.1145/1721837.1721853.
  37. Jiawei Zhang, Bo Chen, and Yinyu Ye. A multiexchange local search algorithm for the capacitated facility location problem. Math. Oper. Res., 30(2):389-403, 2005. URL: https://doi.org/10.1287/MOOR.1040.0125.