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On the Connected Minimum Sum of Radii Problem

Authors Hyung-Chan An , Mong-Jen Kao

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Author Details

Hyung-Chan An
  • Yonsei University, Seoul, Republic of Korea
Mong-Jen Kao
  • National Yang-Ming Chiao-Tung University, Hsinchu, Taiwan

Funding

This work was supported by Institute of Information & communications Technology Planning & Evaluation (IITP) grant funded by the Korea government(MSIT) (No. RS-2021-II212068, Artificial Intelligence Innovation Hub). This work was supported by the National Science and Technology Council (NSTC), Taiwan, grants 112-2628-E-A49-017-MY3, 113-2628-E-A49-017-MY3, and 113-2634-F-A49-001-MBK.

Acknowledgements

The authors made equal contributions. We thank the anonymous reviewers for their helpful comments.

Cite As Get BibTex

Hyung-Chan An and Mong-Jen Kao. On the Connected Minimum Sum of Radii Problem. In 35th International Symposium on Algorithms and Computation (ISAAC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 322, pp. 7:1-7:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.ISAAC.2024.7

Abstract

In this paper, we consider the study for the connected minimum sum of radii problem. In this problem, we are given as input a metric defined on a set of facilities and clients, along with some cost parameters. The objective is to open a subset of facilities, assign every client to an open facilitiy, and connect open facilities using a Steiner tree so that the weighted (by cost parameters) sum of the maximum assignment distance of each facility and the Steiner tree cost is minimized. This problem introduces the min-sum radii objective, an objective function that is widely considered in the clustering literature, to the connected facility location problem, a well-studied network design/clustering problem. This problem is useful in communication network design on a shared medium, or energy optimization of mobile wireless chargers.
We present both a constant-factor approximation algorithm and hardness results for this problem. Our algorithm is based on rounding an LP relaxation that jointly models the min-sum of radii problem and the rooted Steiner tree problem. To round the solution we use a careful clustering procedure that guarantees that every open facility has a proxy client nearby. This allows a reinterpretation for part of the LP solution as a fractional rooted Steiner tree. Combined with a cost filtering technique, this yields a 5.542-approximation algorithm.

Subject Classification

ACM Subject Classification
  • Theory of computation → Facility location and clustering
Keywords
  • connected minimum sum of radii
  • minimum sum of radii
  • connected facility location
  • approximation algorithms
  • Steiner trees

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References

  1. Matthew Andrews and Lisa Zhang. The access network design problem. In 39th Annual Symposium on Foundations of Computer Science, FOCS '98, November 8-11, 1998, Palo Alto, California, USA, pages 40-59. IEEE Computer Society, 1998. URL: https://doi.org/10.1109/SFCS.1998.743427.
  2. Moritz Buchem, Katja Ettmayr, Hugo K. K. Rosado, and Andreas Wiese. A (3 + ε)-approximation algorithm for the minimum sum of radii problem with outliers and extensions for generalized lower bounds. In David P. Woodruff, editor, Proceedings of the 2024 ACM-SIAM Symposium on Discrete Algorithms, SODA 2024, Alexandria, VA, USA, January 7-10, 2024, pages 1738-1765. SIAM, 2024. URL: https://doi.org/10.1137/1.9781611977912.69.
  3. Moses Charikar and Rina Panigrahy. Clustering to minimize the sum of cluster diameters. In Jeffrey Scott Vitter, Paul G. Spirakis, and Mihalis Yannakakis, editors, Proceedings on 33rd Annual ACM Symposium on Theory of Computing, July 6-8, 2001, Heraklion, Crete, Greece, pages 1-10. ACM, 2001. URL: https://doi.org/10.1145/380752.380753.
  4. 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.
  5. Stephen A. Cook. The complexity of theorem-proving procedures. In Michael A. Harrison, Ranan B. Banerji, and Jeffrey D. Ullman, editors, Proceedings of the 3rd Annual ACM Symposium on Theory of Computing, May 3-5, 1971, Shaker Heights, Ohio, USA, pages 151-158. ACM, 1971. URL: https://doi.org/10.1145/800157.805047.
  6. Srinivas Doddi, Madhav V. Marathe, S. S. Ravi, David Scot Taylor, and Peter Widmayer. Approximation algorithms for clustering to minimize the sum of diameters. In Magnús M. Halldórsson, editor, Algorithm Theory - SWAT 2000, 7th Scandinavian Workshop on Algorithm Theory, Bergen, Norway, July 5-7, 2000, Proceedings, volume 1851 of Lecture Notes in Computer Science, pages 237-250. Springer, 2000. URL: https://doi.org/10.1007/3-540-44985-X_22.
  7. Ding-Zhu Du and Peng-Jun Wan. Connected Dominating Set: Theory and Applications. Springer Publishing Company, Incorporated, 2012. Google Scholar
  8. Friedrich Eisenbrand, Fabrizio Grandoni, Thomas Rothvoß, and Guido Schäfer. Approximating connected facility location problems via random facility sampling and core detouring. In Shang-Hua Teng, editor, Proceedings of the Nineteenth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2008, San Francisco, California, USA, January 20-22, 2008, pages 1174-1183. SIAM, 2008. URL: http://dl.acm.org/citation.cfm?id=1347082.1347210.
  9. Zachary Friggstad and Mahya Jamshidian. Improved polynomial-time approximations for clustering with minimum sum of radii or diameters. In Shiri Chechik, Gonzalo Navarro, Eva Rotenberg, and Grzegorz Herman, editors, 30th Annual European Symposium on Algorithms, ESA 2022, September 5-9, 2022, Berlin/Potsdam, Germany, volume 244 of LIPIcs, pages 56:1-56:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPICS.ESA.2022.56.
  10. Zachary Friggstad, Mohsen Rezapour, and Mohammad R. Salavatipour. Approximating connected facility location with lower and upper bounds via LP rounding. In Rasmus Pagh, editor, 15th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2016, June 22-24, 2016, Reykjavik, Iceland, volume 53 of LIPIcs, pages 1:1-1:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. URL: https://doi.org/10.4230/LIPICS.SWAT.2016.1.
  11. Matt Gibson, Gaurav Kanade, Erik Krohn, Imran A. Pirwani, and Kasturi R. Varadarajan. On metric clustering to minimize the sum of radii. Algorithmica, 57(3):484-498, 2010. URL: https://doi.org/10.1007/S00453-009-9282-7.
  12. Michel X. Goemans and David P. Williamson. A general approximation technique for constrained forest problems. SIAM J. Comput., 24(2):296-317, 1995. URL: https://doi.org/10.1137/S0097539793242618.
  13. Sudipto Guha and Samir Khuller. Approximation algorithms for connected dominating sets. Algorithmica, 20(4):374-387, 1998. URL: https://doi.org/10.1007/PL00009201.
  14. Anupam Gupta, Jon M. Kleinberg, Amit Kumar, Rajeev Rastogi, and Bülent Yener. Provisioning a virtual private network: a network design problem for multicommodity flow. In Jeffrey Scott Vitter, Paul G. Spirakis, and Mihalis Yannakakis, editors, Proceedings on 33rd Annual ACM Symposium on Theory of Computing, July 6-8, 2001, Heraklion, Crete, Greece, pages 389-398. ACM, 2001. URL: https://doi.org/10.1145/380752.380830.
  15. Anupam Gupta, Amit Kumar, and Tim Roughgarden. Simpler and better approximation algorithms for network design. In Lawrence L. Larmore and Michel X. Goemans, editors, Proceedings of the 35th Annual ACM Symposium on Theory of Computing, June 9-11, 2003, San Diego, CA, USA, pages 365-372. ACM, 2003. URL: https://doi.org/10.1145/780542.780597.
  16. Pierre Hansen and Brigitte Jaumard. Cluster analysis and mathematical programming. Math. Program., 79:191-215, 1997. URL: https://doi.org/10.1007/BF02614317.
  17. D. S. Hochbaum and D. B. Shmoys. A best possible heuristic for the k-center problem. Mathematics of Operations Research, 10:180-184, 1985. URL: https://doi.org/10.1287/MOOR.10.2.180.
  18. Kamal Jain. A factor 2 approximation algorithm for the generalized steiner network problem. Comb., 21(1):39-60, 2001. URL: https://doi.org/10.1007/S004930170004.
  19. Riheng Jia, Jinhao Wu, Jianfeng Lu, Minglu Li, Feilong Lin, and Zhonglong Zheng. Energy saving in heterogeneous wireless rechargeable sensor networks. In IEEE INFOCOM 2022 - IEEE Conference on Computer Communications, London, United Kingdom, May 2-5, 2022, pages 1838-1847. IEEE, 2022. URL: https://doi.org/10.1109/INFOCOM48880.2022.9796770.
  20. Richard M. Karp. Reducibility among combinatorial problems. In Raymond E. Miller and James W. Thatcher, editors, Proceedings of a symposium on the Complexity of Computer Computations, held March 20-22, 1972, at the IBM Thomas J. Watson Research Center, Yorktown Heights, New York, USA, The IBM Research Symposia Series, pages 85-103. Plenum Press, New York, 1972. URL: https://doi.org/10.1007/978-1-4684-2001-2_9.
  21. Samir Khuller, Manish Purohit, and Kanthi K. Sarpatwar. Analyzing the optimal neighborhood: Algorithms for budgeted and partial connected dominating set problems. In Chandra Chekuri, editor, Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA 2014, Portland, Oregon, USA, January 5-7, 2014, pages 1702-1713. SIAM, 2014. URL: https://doi.org/10.1137/1.9781611973402.123.
  22. Samir Khuller and Sheng Yang. Revisiting connected dominating sets: An almost optimal local information algorithm. Algorithmica, 81(6):2592-2605, 2019. URL: https://doi.org/10.1007/S00453-019-00545-0.
  23. Chaitanya Swamy and Amit Kumar. Primal-dual algorithms for connected facility location problems. Algorithmica, 40(4):245-269, 2004. URL: https://doi.org/10.1007/S00453-004-1112-3.
  24. Wenzheng Xu, Weifa Liang, Xiaohua Jia, Haibin Kan, Yinlong Xu, and Xinming Zhang. Minimizing the maximum charging delay of multiple mobile chargers under the multi-node energy charging scheme. IEEE Trans. Mob. Comput., 20(5):1846-1861, 2021. URL: https://doi.org/10.1109/TMC.2020.2973979.
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